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Properties

Label 2-637-91.89-c1-0-30
Degree $2$
Conductor $637$
Sign $-0.815 + 0.579i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.984i)2-s + (1.25 + 0.724i)3-s − 0.0619i·4-s + (0.172 + 0.643i)5-s + (−0.521 − 1.94i)6-s + (−2.02 + 2.02i)8-s + (−0.450 − 0.780i)9-s + (0.463 − 0.802i)10-s + (−1.24 − 4.65i)11-s + (0.0448 − 0.0776i)12-s + (−3.60 − 0.0282i)13-s + (−0.249 + 0.931i)15-s + 3.87·16-s − 0.467·17-s + (−0.324 + 1.21i)18-s + (−3.26 − 0.873i)19-s + ⋯
L(s)  = 1  + (−0.696 − 0.696i)2-s + (0.724 + 0.418i)3-s − 0.0309i·4-s + (0.0770 + 0.287i)5-s + (−0.213 − 0.795i)6-s + (−0.717 + 0.717i)8-s + (−0.150 − 0.260i)9-s + (0.146 − 0.253i)10-s + (−0.376 − 1.40i)11-s + (0.0129 − 0.0224i)12-s + (−0.999 − 0.00782i)13-s + (−0.0644 + 0.240i)15-s + 0.968·16-s − 0.113·17-s + (−0.0765 + 0.285i)18-s + (−0.748 − 0.200i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.815 + 0.579i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.815 + 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.249251 - 0.781127i\)
\(L(\frac12)\) \(\approx\) \(0.249251 - 0.781127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (3.60 + 0.0282i)T \)
good2 \( 1 + (0.984 + 0.984i)T + 2iT^{2} \)
3 \( 1 + (-1.25 - 0.724i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.172 - 0.643i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.24 + 4.65i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 + (3.26 + 0.873i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.95iT - 23T^{2} \)
29 \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.10 - 1.09i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.38 - 2.38i)T - 37iT^{2} \)
41 \( 1 + (3.68 + 0.986i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.64 + 2.58i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.20 - 3.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 + 4.33i)T + 59iT^{2} \)
61 \( 1 + (4.21 - 2.43i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.03 - 2.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.19 + 0.857i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.0301 + 0.112i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.194 + 0.337i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.5 - 11.5i)T - 83iT^{2} \)
89 \( 1 + (6.83 + 6.83i)T + 89iT^{2} \)
97 \( 1 + (-4.61 - 17.2i)T + (-84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.36524708633927510064312221227, −9.355558528001230269385702576114, −8.675168900794363094061234480907, −8.143593348561141491042782240355, −6.63201875428098287541240653262, −5.73437441033674389188293632336, −4.42289967910256537039342912617, −2.98810461496947495206818632058, −2.47073401704798103673946901119, −0.48449802388070061252783399966, 1.89732778050847488117307685273, 3.05905899326769961348763633700, 4.51703840388458717269905652480, 5.62878211270339646677864738527, 7.17299476398351708401590786516, 7.29993312109189452011815904709, 8.230763606743691315038153414523, 9.043694597309345306967397010328, 9.671688791645941371372569697537, 10.63504834749226516768866771421

Graph of the $Z$-function along the critical line