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Properties

Label 2-845-65.47-c1-0-66
Degree 22
Conductor 845845
Sign −0.0655−0.997i-0.0655 - 0.997i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  − 2.08i·2-s + (1.94 − 1.94i)3-s − 2.35·4-s + (−2.22 + 0.194i)5-s + (−4.06 − 4.06i)6-s − 2.91·7-s + 0.750i·8-s − 4.59i·9-s + (0.405 + 4.65i)10-s + (−0.0186 + 0.0186i)11-s + (−4.59 + 4.59i)12-s + 6.07i·14-s + (−3.96 + 4.71i)15-s − 3.15·16-s + (−2.02 + 2.02i)17-s − 9.58·18-s + ⋯
L(s)  = 1  − 1.47i·2-s + (1.12 − 1.12i)3-s − 1.17·4-s + (−0.996 + 0.0869i)5-s + (−1.66 − 1.66i)6-s − 1.10·7-s + 0.265i·8-s − 1.53i·9-s + (0.128 + 1.47i)10-s + (−0.00561 + 0.00561i)11-s + (−1.32 + 1.32i)12-s + 1.62i·14-s + (−1.02 + 1.21i)15-s − 0.787·16-s + (−0.491 + 0.491i)17-s − 2.26·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.0655−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0655 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.0655−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0655 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.0655−0.997i-0.0655 - 0.997i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(437,⋅)\chi_{845} (437, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.0655−0.997i)(2,\ 845,\ (\ :1/2),\ -0.0655 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.732269+0.781929i0.732269 + 0.781929i
L(12)L(\frac12) ≈\approx 0.732269+0.781929i0.732269 + 0.781929i
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)
bad5 1+(2.22−0.194i)T 1 + (2.22 - 0.194i)T
13 1 1
good2 1+2.08iT−2T2 1 + 2.08iT - 2T^{2}
3 1+(−1.94+1.94i)T−3iT2 1 + (-1.94 + 1.94i)T - 3iT^{2}
7 1+2.91T+7T2 1 + 2.91T + 7T^{2}
11 1+(0.0186−0.0186i)T−11iT2 1 + (0.0186 - 0.0186i)T - 11iT^{2}
17 1+(2.02−2.02i)T−17iT2 1 + (2.02 - 2.02i)T - 17iT^{2}
19 1+(−3.38+3.38i)T−19iT2 1 + (-3.38 + 3.38i)T - 19iT^{2}
23 1+(0.262+0.262i)T+23iT2 1 + (0.262 + 0.262i)T + 23iT^{2}
29 1+4.18iT−29T2 1 + 4.18iT - 29T^{2}
31 1+(−0.835−0.835i)T+31iT2 1 + (-0.835 - 0.835i)T + 31iT^{2}
37 1+6.45T+37T2 1 + 6.45T + 37T^{2}
41 1+(−5.54−5.54i)T+41iT2 1 + (-5.54 - 5.54i)T + 41iT^{2}
43 1+(4.90+4.90i)T+43iT2 1 + (4.90 + 4.90i)T + 43iT^{2}
47 1+0.833T+47T2 1 + 0.833T + 47T^{2}
53 1+(−0.902+0.902i)T−53iT2 1 + (-0.902 + 0.902i)T - 53iT^{2}
59 1+(1.05+1.05i)T+59iT2 1 + (1.05 + 1.05i)T + 59iT^{2}
61 1+10.7T+61T2 1 + 10.7T + 61T^{2}
67 1+12.3iT−67T2 1 + 12.3iT - 67T^{2}
71 1+(2.61+2.61i)T+71iT2 1 + (2.61 + 2.61i)T + 71iT^{2}
73 1+15.0iT−73T2 1 + 15.0iT - 73T^{2}
79 1+4.25iT−79T2 1 + 4.25iT - 79T^{2}
83 1+1.31T+83T2 1 + 1.31T + 83T^{2}
89 1+(−2.36−2.36i)T+89iT2 1 + (-2.36 - 2.36i)T + 89iT^{2}
97 1−0.405iT−97T2 1 - 0.405iT - 97T^{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

−9.496757245427727631920213794136, −8.914426104480271333892641542578, −8.008731967351940467460694177640, −7.12380136372620376527387899503, −6.43887890974344290031804953918, −4.49606352620722985693977297314, −3.34579974555909384345311007536, −3.02513249673087341552977397984, −1.85493424230736158535342335382, −0.44183483362029277417777039296, 2.87857128694904635855283212973, 3.71994041221867545064884238097, 4.56922681398246856320456130132, 5.53962773450378259929843590587, 6.75741485986178006345558998782, 7.46731785990779861463745145952, 8.290287984373353800888332790993, 8.932854343879080968375020325519, 9.565677910872012147001581954389, 10.45225572018238862300459105167

Graph of the ZZ-function along the critical line