Properties

Label 1-967-967.179-r0-0-0
Degree 11
Conductor 967967
Sign 0.0366+0.999i-0.0366 + 0.999i
Analytic cond. 4.490724.49072
Root an. cond. 4.490724.49072
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯

Functional equation

Λ(s)=(967s/2ΓR(s)L(s)=((0.0366+0.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(967s/2ΓR(s)L(s)=((0.0366+0.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0366 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 967967
Sign: 0.0366+0.999i-0.0366 + 0.999i
Analytic conductor: 4.490724.49072
Root analytic conductor: 4.490724.49072
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ967(179,)\chi_{967} (179, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 967, (0: ), 0.0366+0.999i)(1,\ 967,\ (0:\ ),\ -0.0366 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.5673300155+0.5885192417i0.5673300155 + 0.5885192417i
L(12)L(\frac12) \approx 0.5673300155+0.5885192417i0.5673300155 + 0.5885192417i
L(1)L(1) \approx 0.7808924509+0.06885533600i0.7808924509 + 0.06885533600i
L(1)L(1) \approx 0.7808924509+0.06885533600i0.7808924509 + 0.06885533600i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad967 1 1
good2 1+(0.9860.161i)T 1 + (-0.986 - 0.161i)T
3 1+(0.4250.905i)T 1 + (0.425 - 0.905i)T
5 1+(0.581+0.813i)T 1 + (0.581 + 0.813i)T
7 1+(0.701+0.712i)T 1 + (0.701 + 0.712i)T
11 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
13 1+(0.961+0.276i)T 1 + (-0.961 + 0.276i)T
17 1+(0.107+0.994i)T 1 + (-0.107 + 0.994i)T
19 1+(0.927+0.374i)T 1 + (-0.927 + 0.374i)T
23 1+(0.9840.174i)T 1 + (-0.984 - 0.174i)T
29 1+(0.6080.793i)T 1 + (-0.608 - 0.793i)T
31 1+(0.4650.884i)T 1 + (-0.465 - 0.884i)T
37 1+(0.483+0.875i)T 1 + (0.483 + 0.875i)T
41 1+(0.6820.730i)T 1 + (0.682 - 0.730i)T
43 1+(0.254+0.967i)T 1 + (0.254 + 0.967i)T
47 1+(0.0422+0.999i)T 1 + (-0.0422 + 0.999i)T
53 1+(0.247+0.968i)T 1 + (-0.247 + 0.968i)T
59 1+(0.889+0.457i)T 1 + (-0.889 + 0.457i)T
61 1+(0.9740.225i)T 1 + (-0.974 - 0.225i)T
67 1+(0.9990.0390i)T 1 + (0.999 - 0.0390i)T
71 1+(0.3530.935i)T 1 + (0.353 - 0.935i)T
73 1+(0.2470.968i)T 1 + (-0.247 - 0.968i)T
79 1+(0.9110.410i)T 1 + (-0.911 - 0.410i)T
83 1+(0.1520.988i)T 1 + (0.152 - 0.988i)T
89 1+(0.5330.845i)T 1 + (-0.533 - 0.845i)T
97 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
show more
show less
   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.50669125149848746768657417886, −20.516112938196375465253234980618, −20.03556936258931140560980641643, −19.4840873799564153556146763836, −18.20622560820780923855054525232, −17.455682775556939963431446482041, −16.70975669906785365366765330213, −16.35975431024418154953051293438, −15.4177280857491492099478863016, −14.34576948341728887906549673720, −14.01434404035271983297833554841, −12.72184062084601399315979836519, −11.51306625357022111413807783013, −10.84785688754308496940060377049, −10.0479595088366263912648759117, −9.33141479612008227609706780229, −8.631819902777221100819110121412, −7.96664901674456330065567673814, −6.97554030766417845796796622909, −5.6475428250612234223792761145, −5.0292597349267930510552672779, −3.94389415109873394120707315859, −2.66572205261691285622720974500, −1.74827981053160497232937668648, −0.41078189771846528588792989008, 1.73209242325006347007664312388, 2.01912479504055884902578147107, 2.79468727966165255743692454918, 4.21774903482119067220351117500, 6.01352072582362418231128683313, 6.32147610151907112955129794951, 7.597879862549062760087605204013, 7.818015165129935626386779999947, 9.039433279833685621955422676842, 9.62745350269793614095689469430, 10.584457960541144371548544149475, 11.51801819404828420990435257518, 12.26352054501542777434143206564, 12.9083201599976211387646677448, 14.27779625502689334973755781966, 14.85105927518965421543706952914, 15.32522747119038339760657525429, 17.07936553384692470485683628365, 17.32736902548086831822555204995, 18.153676089954569059573101017212, 18.72791223892787792838013920473, 19.35490383732304853171586056808, 20.17053392379486256713224586887, 21.02398006601234558346448589913, 21.7433370491279344804571743562

Graph of the ZZ-function along the critical line