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Properties

Label 2-637-91.4-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.299 - 0.954i$
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  − 0.823i·2-s + (1.33 + 2.30i)3-s + 1.32·4-s + (−2.73 + 1.58i)5-s + (1.89 − 1.09i)6-s − 2.73i·8-s + (−2.03 + 3.53i)9-s + (1.30 + 2.25i)10-s + (−5.14 + 2.97i)11-s + (1.75 + 3.04i)12-s + (0.0766 + 3.60i)13-s + (−7.28 − 4.20i)15-s + 0.390·16-s − 2.69·17-s + (2.90 + 1.67i)18-s + (1.69 + 0.978i)19-s + ⋯
L(s)  = 1  − 0.582i·2-s + (0.767 + 1.33i)3-s + 0.660·4-s + (−1.22 + 0.707i)5-s + (0.774 − 0.447i)6-s − 0.967i·8-s + (−0.679 + 1.17i)9-s + (0.411 + 0.713i)10-s + (−1.55 + 0.895i)11-s + (0.507 + 0.879i)12-s + (0.0212 + 0.999i)13-s + (−1.88 − 1.08i)15-s + 0.0976·16-s − 0.654·17-s + (0.685 + 0.395i)18-s + (0.388 + 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.299 - 0.954i$
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: \((2,\ 637,\ (\ :1/2),\ -0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889918 + 1.21209i\)
\(L(\frac12)\) \(\approx\) \(0.889918 + 1.21209i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0766 - 3.60i)T \)
good2 \( 1 + 0.823iT - 2T^{2} \)
3 \( 1 + (-1.33 - 2.30i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.73 - 1.58i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.14 - 2.97i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 + (-1.69 - 0.978i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.997 - 0.575i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.50iT - 37T^{2} \)
41 \( 1 + (-3.23 - 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.49 - 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.394 + 0.228i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.199 - 0.345i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.80iT - 59T^{2} \)
61 \( 1 + (0.578 - 1.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.43 + 3.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.90 + 2.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.19 + 4.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.95 - 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.19iT - 83T^{2} \)
89 \( 1 - 3.56iT - 89T^{2} \)
97 \( 1 + (-2.96 + 1.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.82079884594473709387254095606, −10.09563009452748901431733278778, −9.447776072808326725678789720311, −8.184464151505936009255913926174, −7.51466693209386793035668543423, −6.57684424436489311073584729920, −4.80678254286645028307165682800, −4.09016843384854143067181245782, −3.10627473719498808888576967346, −2.40885385650061136543442387225, 0.71111397246126526966327570571, 2.45010953959591611420510335294, 3.27655887040785924003661459237, 5.02936967835365309221759360656, 5.95663458703174565100471480946, 7.32091847360063130320425239285, 7.47988354001781969962721326097, 8.435273193380735277719271514780, 8.645688502965525237835995001315, 10.59536069997080740596095449124

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