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Properties

Label 2-637-91.41-c1-0-35
Degree $2$
Conductor $637$
Sign $0.971 - 0.237i$
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  + (2.35 + 0.629i)2-s + (1.70 − 0.985i)3-s + (3.39 + 1.96i)4-s + (−0.166 − 0.166i)5-s + (4.63 − 1.24i)6-s + (3.30 + 3.30i)8-s + (0.442 − 0.766i)9-s + (−0.286 − 0.495i)10-s + (−0.740 + 2.76i)11-s + 7.72·12-s + (−2.03 − 2.97i)13-s + (−0.447 − 0.119i)15-s + (1.76 + 3.05i)16-s + (2.01 − 3.49i)17-s + (1.52 − 1.52i)18-s + (−5.30 + 1.42i)19-s + ⋯
L(s)  = 1  + (1.66 + 0.445i)2-s + (0.985 − 0.568i)3-s + (1.69 + 0.980i)4-s + (−0.0743 − 0.0743i)5-s + (1.89 − 0.506i)6-s + (1.16 + 1.16i)8-s + (0.147 − 0.255i)9-s + (−0.0904 − 0.156i)10-s + (−0.223 + 0.833i)11-s + 2.23·12-s + (−0.563 − 0.826i)13-s + (−0.115 − 0.0309i)15-s + (0.441 + 0.764i)16-s + (0.488 − 0.846i)17-s + (0.358 − 0.358i)18-s + (−1.21 + 0.326i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.63014 + 0.556970i\)
\(L(\frac12)\) \(\approx\) \(4.63014 + 0.556970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 2.97i)T \)
good2 \( 1 + (-2.35 - 0.629i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-1.70 + 0.985i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.166 + 0.166i)T + 5iT^{2} \)
11 \( 1 + (0.740 - 2.76i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.01 + 3.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.30 - 1.42i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.23 - 3.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.62 - 1.62i)T + 31iT^{2} \)
37 \( 1 + (-2.68 + 10.0i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.0713 - 0.266i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.91 - 2.25i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.30 - 2.30i)T - 47iT^{2} \)
53 \( 1 - 7.93T + 53T^{2} \)
59 \( 1 + (0.188 + 0.702i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.97 + 1.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.57 + 0.688i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.362 + 1.35i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (9.58 - 9.58i)T - 73iT^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
83 \( 1 + (-4.19 - 4.19i)T + 83iT^{2} \)
89 \( 1 + (3.33 + 0.893i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.61 + 0.701i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.79016476529051053996086415785, −9.760076515043432321204509634510, −8.472123725949248843873296766904, −7.63072327876390803396335373631, −7.11712931861295375440880283375, −5.97067163573722203432846427524, −5.03274606757587068873427865758, −4.09340488547657810086289641329, −2.93616456533778366041182463349, −2.19063998138080432402563987681, 2.14691010878075244716769857469, 3.01998563265648458727754257309, 3.98658024218054189606658466464, 4.53749795505112046616710291814, 5.87209310901013686061822886240, 6.54844630036696069653117923140, 8.008635025045900707478280483333, 8.794856812660075185277386280436, 9.951017095969241662903739493039, 10.65055495119534445138609553195

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