Properties

Label 2-637-13.4-c1-0-17
Degree $2$
Conductor $637$
Sign $0.928 + 0.371i$
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  + (−0.433 − 0.249i)2-s + (0.424 − 0.735i)3-s + (−0.875 − 1.51i)4-s + 1.04i·5-s + (−0.367 + 0.212i)6-s + 1.87i·8-s + (1.13 + 1.97i)9-s + (0.260 − 0.451i)10-s + (3.43 + 1.98i)11-s − 1.48·12-s + (3.57 − 0.468i)13-s + (0.767 + 0.442i)15-s + (−1.28 + 2.21i)16-s + (0.0710 + 0.123i)17-s − 1.13i·18-s + (−4.77 + 2.75i)19-s + ⋯
L(s)  = 1  + (−0.306 − 0.176i)2-s + (0.245 − 0.424i)3-s + (−0.437 − 0.757i)4-s + 0.466i·5-s + (−0.150 + 0.0867i)6-s + 0.662i·8-s + (0.379 + 0.657i)9-s + (0.0824 − 0.142i)10-s + (1.03 + 0.598i)11-s − 0.429·12-s + (0.991 − 0.129i)13-s + (0.198 + 0.114i)15-s + (−0.320 + 0.554i)16-s + (0.0172 + 0.0298i)17-s − 0.268i·18-s + (−1.09 + 0.632i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34680 - 0.259423i\)
\(L(\frac12)\) \(\approx\) \(1.34680 - 0.259423i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.57 + 0.468i)T \)
good2 \( 1 + (0.433 + 0.249i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.424 + 0.735i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.04iT - 5T^{2} \)
11 \( 1 + (-3.43 - 1.98i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0710 - 0.123i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.77 - 2.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 3.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.84iT - 31T^{2} \)
37 \( 1 + (-0.730 - 0.421i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.55iT - 47T^{2} \)
53 \( 1 + 0.279T + 53T^{2} \)
59 \( 1 + (-9.33 + 5.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.93 + 5.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.45 - 2.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.20 - 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.61iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + (-1.51 - 0.873i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.34 - 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.50916704808931074554126237434, −9.778247141834577623176691313946, −8.697243564759855170007766316286, −8.171124153507300753225868538385, −6.79858567001182967976949318487, −6.28967219940246294005879502322, −4.93292919932815786700403208293, −3.98725960937552746534450301818, −2.33163009135923955540045653409, −1.26006995341197000089899129657, 1.06805285349097292043703720864, 3.25521062628946034535734233028, 3.93647851450687564275636576068, 4.87433545253310932541914804012, 6.39557234548306064242533655507, 7.04131205036069099006191247556, 8.513673817894659187413575753326, 8.762603238019113643576958649149, 9.410631019431990812543314959486, 10.50420759839937615913624798785

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