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Properties

Label 2-637-91.4-c1-0-34
Degree $2$
Conductor $637$
Sign $0.248 + 0.968i$
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  − 1.27i·2-s + (0.583 + 1.01i)3-s + 0.370·4-s + (1.57 − 0.907i)5-s + (1.29 − 0.745i)6-s − 3.02i·8-s + (0.817 − 1.41i)9-s + (−1.15 − 2.00i)10-s + (2.40 − 1.38i)11-s + (0.216 + 0.374i)12-s + (−3.58 + 0.402i)13-s + (1.83 + 1.05i)15-s − 3.12·16-s − 2.74·17-s + (−1.80 − 1.04i)18-s + (5.08 + 2.93i)19-s + ⋯
L(s)  = 1  − 0.902i·2-s + (0.337 + 0.583i)3-s + 0.185·4-s + (0.702 − 0.405i)5-s + (0.527 − 0.304i)6-s − 1.06i·8-s + (0.272 − 0.472i)9-s + (−0.366 − 0.634i)10-s + (0.725 − 0.418i)11-s + (0.0624 + 0.108i)12-s + (−0.993 + 0.111i)13-s + (0.473 + 0.273i)15-s − 0.780·16-s − 0.665·17-s + (−0.426 − 0.246i)18-s + (1.16 + 0.673i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.248 + 0.968i$
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: \((2,\ 637,\ (\ :1/2),\ 0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69177 - 1.31225i\)
\(L(\frac12)\) \(\approx\) \(1.69177 - 1.31225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.58 - 0.402i)T \)
good2 \( 1 + 1.27iT - 2T^{2} \)
3 \( 1 + (-0.583 - 1.01i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.57 + 0.907i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 + (-5.08 - 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.79 - 1.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.74iT - 37T^{2} \)
41 \( 1 + (-5.51 - 3.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.76 - 3.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.24 + 9.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.07iT - 59T^{2} \)
61 \( 1 + (0.540 - 0.936i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.34 - 2.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.35 + 1.35i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.64 + 3.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.97iT - 83T^{2} \)
89 \( 1 - 16.0iT - 89T^{2} \)
97 \( 1 + (12.3 - 7.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.14758500809978161935369400094, −9.675098893603315224495017730940, −9.213986117414690032896706938286, −7.897241878085504760715317001944, −6.71849890661787393301478674957, −5.85569094247099518857479155605, −4.47206905984375505757952837862, −3.62648909105687383994464634637, −2.48947337808136539436399652491, −1.26382935065783460890187967880, 1.90199270524595445907021938291, 2.65184053021041713941785789926, 4.52194926579488812877100355296, 5.57987109611810271576829282620, 6.53516315848215209705724632447, 7.18787094301381599276220904040, 7.77636028971408627880900554623, 8.841850498345377555644473114532, 9.827545296614182891342317037824, 10.62173759456994348991830253365

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