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Properties

Label 2-637-91.54-c1-0-36
Degree $2$
Conductor $637$
Sign $-0.333 + 0.942i$
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  + (1.51 − 1.51i)2-s + (−0.0170 − 0.00986i)3-s − 2.58i·4-s + (1.34 − 0.360i)5-s + (−0.0408 + 0.0109i)6-s + (−0.893 − 0.893i)8-s + (−1.49 − 2.59i)9-s + (1.49 − 2.58i)10-s + (0.336 − 0.0902i)11-s + (−0.0255 + 0.0442i)12-s + (1.32 − 3.35i)13-s + (−0.0265 − 0.00711i)15-s + 2.47·16-s + 0.982·17-s + (−6.20 − 1.66i)18-s + (−1.19 + 4.45i)19-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)2-s + (−0.00986 − 0.00569i)3-s − 1.29i·4-s + (0.601 − 0.161i)5-s + (−0.0166 + 0.00446i)6-s + (−0.315 − 0.315i)8-s + (−0.499 − 0.865i)9-s + (0.471 − 0.816i)10-s + (0.101 − 0.0272i)11-s + (−0.00737 + 0.0127i)12-s + (0.368 − 0.929i)13-s + (−0.00685 − 0.00183i)15-s + 0.618·16-s + 0.238·17-s + (−1.46 − 0.392i)18-s + (−0.273 + 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61953 - 2.29026i\)
\(L(\frac12)\) \(\approx\) \(1.61953 - 2.29026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 3.35i)T \)
good2 \( 1 + (-1.51 + 1.51i)T - 2iT^{2} \)
3 \( 1 + (0.0170 + 0.00986i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 0.360i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.336 + 0.0902i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 0.982T + 17T^{2} \)
19 \( 1 + (1.19 - 4.45i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + (0.941 + 1.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.755 + 2.81i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 + (0.580 - 2.16i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.47 + 3.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.83 - 10.5i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.77 - 6.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.7 - 10.7i)T - 59iT^{2} \)
61 \( 1 + (-5.59 + 3.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.61 - 9.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.43 - 9.07i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (10.3 + 2.76i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.890 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.33 + 8.33i)T + 83iT^{2} \)
89 \( 1 + (-9.61 + 9.61i)T - 89iT^{2} \)
97 \( 1 + (-12.6 + 3.39i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.39117557644644582899631486581, −9.866690141324945542759851742726, −8.742684593786881368456185713129, −7.74897269417391949787167949133, −6.05638075362005834402684894586, −5.80326685537278145097648832870, −4.51352361194026142949624169416, −3.53136678117615623011466858267, −2.63186851723546971739544777919, −1.24823596529450157848968796626, 2.07523444314963851908345975609, 3.52188922884700742511390483765, 4.66806145419043410382910374177, 5.42046583248516951711547569782, 6.27057139856806068116893232147, 7.01509074580527713996212918568, 7.940765275707639780099067023338, 8.915999313405176590456548927893, 9.945751201638360122191464939450, 10.97577142420208796630703787758

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