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Properties

Label 2-637-91.45-c1-0-23
Degree $2$
Conductor $637$
Sign $0.999 + 0.0104i$
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  + (1.14 − 1.14i)2-s + (0.445 − 0.256i)3-s − 0.630i·4-s + (−0.395 + 1.47i)5-s + (0.215 − 0.805i)6-s + (1.57 + 1.57i)8-s + (−1.36 + 2.36i)9-s + (1.23 + 2.14i)10-s + (−0.745 + 2.78i)11-s + (−0.162 − 0.280i)12-s + (2.94 − 2.08i)13-s + (0.203 + 0.757i)15-s + 4.86·16-s + 6.21·17-s + (1.14 + 4.28i)18-s + (−2.23 + 0.598i)19-s + ⋯
L(s)  = 1  + (0.811 − 0.811i)2-s + (0.256 − 0.148i)3-s − 0.315i·4-s + (−0.176 + 0.659i)5-s + (0.0880 − 0.328i)6-s + (0.555 + 0.555i)8-s + (−0.455 + 0.789i)9-s + (0.391 + 0.678i)10-s + (−0.224 + 0.838i)11-s + (−0.0467 − 0.0810i)12-s + (0.816 − 0.577i)13-s + (0.0524 + 0.195i)15-s + 1.21·16-s + 1.50·17-s + (0.270 + 1.01i)18-s + (−0.512 + 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.999 + 0.0104i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.999 + 0.0104i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44432 - 0.0127926i\)
\(L(\frac12)\) \(\approx\) \(2.44432 - 0.0127926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.94 + 2.08i)T \)
good2 \( 1 + (-1.14 + 1.14i)T - 2iT^{2} \)
3 \( 1 + (-0.445 + 0.256i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.395 - 1.47i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.745 - 2.78i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 6.21T + 17T^{2} \)
19 \( 1 + (2.23 - 0.598i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 + (-0.379 + 0.656i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.36 - 2.24i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (-1.94 + 0.522i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.24 - 1.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.13 - 0.571i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.47 + 4.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.623 + 0.623i)T - 59iT^{2} \)
61 \( 1 + (4.48 + 2.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.1 - 4.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.3 + 2.76i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.80 + 6.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.24 + 7.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.51 - 1.51i)T + 83iT^{2} \)
89 \( 1 + (-5.91 + 5.91i)T - 89iT^{2} \)
97 \( 1 + (-0.933 + 3.48i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.67854442079104690180312007877, −10.24642813811254956061430974937, −8.728799970356455635634593927084, −7.88570440733454121648123053122, −7.17977883934165704677893943519, −5.75621096555099330187827065276, −4.90635851950369497787728273143, −3.68408891353721038654455403435, −2.91845012283628037002868097581, −1.86748606928768742604690668619, 1.14323863022383582757738097184, 3.35579935755127852404905884709, 4.03000334334685012511015040751, 5.32408407782011008104544106501, 5.85414188998917093424445600276, 6.82140142513130874303220553519, 7.905974973835547659716040642187, 8.725797523803258401307272292121, 9.516732282400287793231997028104, 10.59601938987809657282612813245

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