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Properties

Label 2-637-91.4-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.372 - 0.927i$
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  − 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

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.372 - 0.927i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615027 + 0.910014i\)
\(L(\frac12)\) \(\approx\) \(0.615027 + 0.910014i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + 2.18iT - 2T^{2} \)
3 \( 1 + (0.895 + 1.55i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.89 + 1.09i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.10 + 0.637i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + (5.68 + 3.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + (-1.10 + 1.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + (2.20 + 1.27i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.70 + 2.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.08 + 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.85iT - 59T^{2} \)
61 \( 1 + (-6.37 + 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.87 - 5.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.791 - 0.456i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3 + 1.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.55iT - 83T^{2} \)
89 \( 1 + 2.91iT - 89T^{2} \)
97 \( 1 + (-13.1 + 7.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15930624084380098654332368821, −9.268551201465490186357175743835, −8.732203680607250584980981532085, −7.07829446431214611821892572457, −6.45740792489184069708560120282, −5.16368294417713953081998970209, −4.22720016079006004795848593384, −2.65797190259664534910305746339, −1.78838542180004913324572604439, −0.63068482637995740321499257054, 2.43295887203806432238362271193, 4.32948954678276299851584239643, 4.94130153665576688602015884051, 5.91958148762236628856279437785, 6.56139384277718308642843366832, 7.41797706601794476924588567162, 8.514282798949624798695065891553, 9.390542324136951992824605773174, 10.19885787751744967158820105754, 10.79962877993932160195884515155

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