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Properties

Label 2-637-91.4-c1-0-7
Degree $2$
Conductor $637$
Sign $-0.862 + 0.506i$
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  + 1.34i·2-s + (1.02 + 1.77i)3-s + 0.190·4-s + (−3.08 + 1.78i)5-s + (−2.38 + 1.37i)6-s + 2.94i·8-s + (−0.601 + 1.04i)9-s + (−2.39 − 4.15i)10-s + (−1.10 + 0.639i)11-s + (0.195 + 0.337i)12-s + (−3.57 − 0.474i)13-s + (−6.33 − 3.65i)15-s − 3.58·16-s + 7.73·17-s + (−1.40 − 0.809i)18-s + (−0.817 − 0.471i)19-s + ⋯
L(s)  = 1  + 0.951i·2-s + (0.591 + 1.02i)3-s + 0.0951·4-s + (−1.38 + 0.797i)5-s + (−0.975 + 0.562i)6-s + 1.04i·8-s + (−0.200 + 0.347i)9-s + (−0.758 − 1.31i)10-s + (−0.333 + 0.192i)11-s + (0.0563 + 0.0975i)12-s + (−0.991 − 0.131i)13-s + (−1.63 − 0.944i)15-s − 0.895·16-s + 1.87·17-s + (−0.330 − 0.190i)18-s + (−0.187 − 0.108i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.862 + 0.506i$
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: \((2,\ 637,\ (\ :1/2),\ -0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351789 - 1.29323i\)
\(L(\frac12)\) \(\approx\) \(0.351789 - 1.29323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 + 0.474i)T \)
good2 \( 1 - 1.34iT - 2T^{2} \)
3 \( 1 + (-1.02 - 1.77i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.08 - 1.78i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 - 0.639i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.73T + 17T^{2} \)
19 \( 1 + (0.817 + 0.471i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 + (2.02 - 3.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 + 2.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.05iT - 37T^{2} \)
41 \( 1 + (-3.63 - 2.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 - 3.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.774 - 0.447i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.1iT - 59T^{2} \)
61 \( 1 + (3.81 - 6.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.47 + 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.89 + 5.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.658 + 0.380i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.32iT - 83T^{2} \)
89 \( 1 + 7.57iT - 89T^{2} \)
97 \( 1 + (-0.414 + 0.239i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.90620902542755210754067152754, −10.21395044603882795047395721730, −9.282341847662855939932315018515, −8.091568438268865984055815632327, −7.64656595282411331931851321873, −6.95539651629734244900742967829, −5.64712447789001887589402435702, −4.61335071186408178898067300245, −3.56414454934158352745115032676, −2.73873889293131526472561419446, 0.67118411711015352881383027137, 1.93337787714907901298562946484, 3.12144536135279713519948923074, 4.02245461065273007139935321669, 5.31472012240852477416867877638, 6.87526734139363149174583780730, 7.74464314117782028866217866376, 7.990196483355920001346673964674, 9.231702836668459382215814814665, 10.17217893190561546022976108376

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