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Properties

Label 2-637-91.19-c1-0-37
Degree $2$
Conductor $637$
Sign $0.746 + 0.664i$
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  + (2.26 + 0.607i)2-s − 2.76i·3-s + (3.03 + 1.75i)4-s + (1.53 − 0.412i)5-s + (1.67 − 6.25i)6-s + (2.50 + 2.50i)8-s − 4.61·9-s + 3.74·10-s + (2.22 + 2.22i)11-s + (4.84 − 8.39i)12-s + (1.04 + 3.44i)13-s + (−1.13 − 4.24i)15-s + (0.649 + 1.12i)16-s + (−0.320 + 0.555i)17-s + (−10.4 − 2.80i)18-s + (−5.57 − 5.57i)19-s + ⋯
L(s)  = 1  + (1.60 + 0.429i)2-s − 1.59i·3-s + (1.51 + 0.877i)4-s + (0.688 − 0.184i)5-s + (0.684 − 2.55i)6-s + (0.886 + 0.886i)8-s − 1.53·9-s + 1.18·10-s + (0.669 + 0.669i)11-s + (1.39 − 2.42i)12-s + (0.290 + 0.956i)13-s + (−0.293 − 1.09i)15-s + (0.162 + 0.281i)16-s + (−0.0778 + 0.134i)17-s + (−2.46 − 0.661i)18-s + (−1.27 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.746 + 0.664i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.746 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.62059 - 1.37817i\)
\(L(\frac12)\) \(\approx\) \(3.62059 - 1.37817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 3.44i)T \)
good2 \( 1 + (-2.26 - 0.607i)T + (1.73 + i)T^{2} \)
3 \( 1 + 2.76iT - 3T^{2} \)
5 \( 1 + (-1.53 + 0.412i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
17 \( 1 + (0.320 - 0.555i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.57 + 5.57i)T + 19iT^{2} \)
23 \( 1 + (-0.126 + 0.0730i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 - 6.46i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.00 - 3.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.60 - 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.816 - 3.04i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.66 - 6.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.07 + 4.00i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 4.50iT - 61T^{2} \)
67 \( 1 + (1.00 - 1.00i)T - 67iT^{2} \)
71 \( 1 + (-13.8 - 3.70i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.81 - 1.82i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.316 + 0.548i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.07 + 1.07i)T + 83iT^{2} \)
89 \( 1 + (-13.1 - 3.51i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.0487 - 0.181i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.98600144164652424242685096356, −9.426133875866494760654881454031, −8.472962466438947442883469194969, −7.21937777003033960879595443025, −6.64998644683266624302997000756, −6.21646814424031076948370843478, −5.07402852062439854368010268393, −4.07533123761379967192029540334, −2.54747470454101902111450001104, −1.66677588965230044423959540629, 2.22927494274450746130484849933, 3.53705799619171749677669863861, 3.91232078653303059479931190752, 5.05573026412867257580990005835, 5.78954750675604423646137243779, 6.38987987632455529180334610768, 8.253881562240593247138917991145, 9.248840506841924275737128387825, 10.26324344173923009981795320229, 10.66543834301967453874901075291

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