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Properties

Label 2-156-156.107-c1-0-1
Degree 22
Conductor 156156
Sign −0.318−0.947i-0.318 - 0.947i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 0.254i)2-s + (0.564 + 1.63i)3-s + (1.87 + 0.707i)4-s + 3.40i·5-s + (−0.369 − 2.42i)6-s + (−2.05 − 1.18i)7-s + (−2.42 − 1.45i)8-s + (−2.36 + 1.84i)9-s + (0.865 − 4.73i)10-s + (−1.63 − 2.82i)11-s + (−0.101 + 3.46i)12-s + (2.06 + 2.95i)13-s + (2.56 + 2.17i)14-s + (−5.57 + 1.92i)15-s + (2.99 + 2.64i)16-s + (−0.380 − 0.219i)17-s + ⋯
L(s)  = 1  + (−0.983 − 0.179i)2-s + (0.326 + 0.945i)3-s + (0.935 + 0.353i)4-s + 1.52i·5-s + (−0.150 − 0.988i)6-s + (−0.777 − 0.449i)7-s + (−0.856 − 0.516i)8-s + (−0.787 + 0.616i)9-s + (0.273 − 1.49i)10-s + (−0.492 − 0.852i)11-s + (−0.0293 + 0.999i)12-s + (0.571 + 0.820i)13-s + (0.684 + 0.581i)14-s + (−1.43 + 0.496i)15-s + (0.749 + 0.661i)16-s + (−0.0922 − 0.0532i)17-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((−0.318−0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((−0.318−0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: −0.318−0.947i-0.318 - 0.947i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(107,⋅)\chi_{156} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), −0.318−0.947i)(2,\ 156,\ (\ :1/2),\ -0.318 - 0.947i)

Particular Values

L(1)L(1) ≈\approx 0.414081+0.575960i0.414081 + 0.575960i
L(12)L(\frac12) ≈\approx 0.414081+0.575960i0.414081 + 0.575960i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.39+0.254i)T 1 + (1.39 + 0.254i)T
3 1+(−0.564−1.63i)T 1 + (-0.564 - 1.63i)T
13 1+(−2.06−2.95i)T 1 + (-2.06 - 2.95i)T
good5 1−3.40iT−5T2 1 - 3.40iT - 5T^{2}
7 1+(2.05+1.18i)T+(3.5+6.06i)T2 1 + (2.05 + 1.18i)T + (3.5 + 6.06i)T^{2}
11 1+(1.63+2.82i)T+(−5.5+9.52i)T2 1 + (1.63 + 2.82i)T + (-5.5 + 9.52i)T^{2}
17 1+(0.380+0.219i)T+(8.5+14.7i)T2 1 + (0.380 + 0.219i)T + (8.5 + 14.7i)T^{2}
19 1+(−2.98−1.72i)T+(9.5+16.4i)T2 1 + (-2.98 - 1.72i)T + (9.5 + 16.4i)T^{2}
23 1+(−3.46−6.00i)T+(−11.5+19.9i)T2 1 + (-3.46 - 6.00i)T + (-11.5 + 19.9i)T^{2}
29 1+(−7.57+4.37i)T+(14.5−25.1i)T2 1 + (-7.57 + 4.37i)T + (14.5 - 25.1i)T^{2}
31 1−0.323iT−31T2 1 - 0.323iT - 31T^{2}
37 1+(−1.58−2.74i)T+(−18.5+32.0i)T2 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.42+0.823i)T+(20.5−35.5i)T2 1 + (-1.42 + 0.823i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.845−0.488i)T+(21.5+37.2i)T2 1 + (-0.845 - 0.488i)T + (21.5 + 37.2i)T^{2}
47 1−2.05T+47T2 1 - 2.05T + 47T^{2}
53 1−5.71iT−53T2 1 - 5.71iT - 53T^{2}
59 1+(−2.43+4.21i)T+(−29.5−51.0i)T2 1 + (-2.43 + 4.21i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6.60+11.4i)T+(−30.5−52.8i)T2 1 + (-6.60 + 11.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.51−2.03i)T+(33.5−58.0i)T2 1 + (3.51 - 2.03i)T + (33.5 - 58.0i)T^{2}
71 1+(−0.174+0.302i)T+(−35.5−61.4i)T2 1 + (-0.174 + 0.302i)T + (-35.5 - 61.4i)T^{2}
73 1+5.87T+73T2 1 + 5.87T + 73T^{2}
79 1−13.0iT−79T2 1 - 13.0iT - 79T^{2}
83 1+1.55T+83T2 1 + 1.55T + 83T^{2}
89 1+(−10.7+6.19i)T+(44.5−77.0i)T2 1 + (-10.7 + 6.19i)T + (44.5 - 77.0i)T^{2}
97 1+(−0.991+1.71i)T+(−48.5−84.0i)T2 1 + (-0.991 + 1.71i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.50132933006216811539581313465, −11.58243899290019791221875384181, −10.97229735445079802177858407475, −10.15772456876450968452991807603, −9.458229287505415245878451995668, −8.205700659353552018685369355694, −7.04750366312889527530628247538, −6.02124092471671874435601580865, −3.60513390079610943032977185867, −2.84964818474961997325598018704, 0.932436248914981498181819188417, 2.70126799880185851266631645857, 5.23375251624566584066562585352, 6.42972377404300305736652060113, 7.61899150218481839798599385319, 8.628521201518085571955480566679, 9.133836141034799932135789775201, 10.36470438088096865809809620446, 11.91124749900907373354332538119, 12.63667238517546045644213749631

Graph of the ZZ-function along the critical line