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Properties

Label 2-637-91.23-c1-0-14
Degree 22
Conductor 637637
Sign 0.372−0.927i0.372 - 0.927i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
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  + 2.18i·2-s + (0.895 − 1.55i)3-s − 2.79·4-s + (−1.89 − 1.09i)5-s + (3.39 + 1.96i)6-s − 1.73i·8-s + (−0.104 − 0.180i)9-s + (2.39 − 4.14i)10-s + (1.10 + 0.637i)11-s + (−2.49 + 4.33i)12-s + (3.5 + 0.866i)13-s + (−3.39 + 1.96i)15-s − 1.79·16-s + 3·17-s + (0.395 − 0.228i)18-s + (5.68 − 3.28i)19-s + ⋯
L(s)  = 1  + 1.54i·2-s + (0.517 − 0.895i)3-s − 1.39·4-s + (−0.847 − 0.489i)5-s + (1.38 + 0.800i)6-s − 0.612i·8-s + (−0.0347 − 0.0602i)9-s + (0.757 − 1.31i)10-s + (0.332 + 0.192i)11-s + (−0.721 + 1.24i)12-s + (0.970 + 0.240i)13-s + (−0.876 + 0.506i)15-s − 0.447·16-s + 0.727·17-s + (0.0932 − 0.0538i)18-s + (1.30 − 0.753i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.372−0.927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.372−0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.372−0.927i0.372 - 0.927i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(569,⋅)\chi_{637} (569, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.372−0.927i)(2,\ 637,\ (\ :1/2),\ 0.372 - 0.927i)

Particular Values

L(1)L(1) ≈\approx 1.33406+0.901622i1.33406 + 0.901622i
L(12)L(\frac12) ≈\approx 1.33406+0.901622i1.33406 + 0.901622i
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)
bad7 1 1
13 1+(−3.5−0.866i)T 1 + (-3.5 - 0.866i)T
good2 1−2.18iT−2T2 1 - 2.18iT - 2T^{2}
3 1+(−0.895+1.55i)T+(−1.5−2.59i)T2 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.89+1.09i)T+(2.5+4.33i)T2 1 + (1.89 + 1.09i)T + (2.5 + 4.33i)T^{2}
11 1+(−1.10−0.637i)T+(5.5+9.52i)T2 1 + (-1.10 - 0.637i)T + (5.5 + 9.52i)T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+(−5.68+3.28i)T+(9.5−16.4i)T2 1 + (-5.68 + 3.28i)T + (9.5 - 16.4i)T^{2}
23 1−7.58T+23T2 1 - 7.58T + 23T^{2}
29 1+(−1.10−1.91i)T+(−14.5+25.1i)T2 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2}
31 1+(7.5−4.33i)T+(15.5−26.8i)T2 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2}
37 1−6.92iT−37T2 1 - 6.92iT - 37T^{2}
41 1+(−2.20+1.27i)T+(20.5−35.5i)T2 1 + (-2.20 + 1.27i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.18+3.78i)T+(−21.5−37.2i)T2 1 + (-2.18 + 3.78i)T + (-21.5 - 37.2i)T^{2}
47 1+(3.70+2.14i)T+(23.5+40.7i)T2 1 + (3.70 + 2.14i)T + (23.5 + 40.7i)T^{2}
53 1+(−6.08−10.5i)T+(−26.5+45.8i)T2 1 + (-6.08 - 10.5i)T + (-26.5 + 45.8i)T^{2}
59 1−8.85iT−59T2 1 - 8.85iT - 59T^{2}
61 1+(6.37+11.0i)T+(−30.5+52.8i)T2 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(9.87+5.70i)T+(33.5+58.0i)T2 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2}
71 1+(0.791+0.456i)T+(35.5+61.4i)T2 1 + (0.791 + 0.456i)T + (35.5 + 61.4i)T^{2}
73 1+(−3+1.73i)T+(36.5−63.2i)T2 1 + (-3 + 1.73i)T + (36.5 - 63.2i)T^{2}
79 1+(−3+5.19i)T+(−39.5−68.4i)T2 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2}
83 1+3.55iT−83T2 1 + 3.55iT - 83T^{2}
89 1+2.91iT−89T2 1 + 2.91iT - 89T^{2}
97 1+(13.1+7.61i)T+(48.5+84.0i)T2 1 + (13.1 + 7.61i)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

−10.79526417684735664833950029497, −9.159331419569469170663478863689, −8.735078549943441784494869676900, −7.81721471721968327102241989906, −7.31145815459036313551312016574, −6.64517179320774365373490598418, −5.42908137548725399376775896928, −4.56578541770108484399372854762, −3.19851392988863822957567174270, −1.22207996422282042602728488688, 1.14960984070049545214224612595, 2.97442661593573578698350622305, 3.56650979106340368876028914644, 4.10882791166648357287218098820, 5.49956221424824451192111860178, 7.04585765583933580504961077853, 8.105569118275456025999076142019, 9.192701300471768681622905416372, 9.616816874499505150984702890608, 10.61076657448359154929199690277

Graph of the ZZ-function along the critical line