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Properties

Label 2-637-7.4-c1-0-26
Degree $2$
Conductor $637$
Sign $-0.947 + 0.318i$
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  + (−1.09 − 1.89i)2-s + (0.879 − 1.52i)3-s + (−1.38 + 2.39i)4-s + (1.05 + 1.82i)5-s − 3.83·6-s + 1.67·8-s + (−0.0460 − 0.0797i)9-s + (2.30 − 3.99i)10-s + (2.88 − 4.99i)11-s + (2.43 + 4.21i)12-s − 13-s + 3.71·15-s + (0.939 + 1.62i)16-s + (0.820 − 1.42i)17-s + (−0.100 + 0.174i)18-s + (−1.33 − 2.31i)19-s + ⋯
L(s)  = 1  + (−0.771 − 1.33i)2-s + (0.507 − 0.879i)3-s + (−0.691 + 1.19i)4-s + (0.471 + 0.817i)5-s − 1.56·6-s + 0.591·8-s + (−0.0153 − 0.0265i)9-s + (0.728 − 1.26i)10-s + (0.869 − 1.50i)11-s + (0.702 + 1.21i)12-s − 0.277·13-s + 0.958·15-s + (0.234 + 0.406i)16-s + (0.198 − 0.344i)17-s + (−0.0236 + 0.0410i)18-s + (−0.306 − 0.530i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194151 - 1.18713i\)
\(L(\frac12)\) \(\approx\) \(0.194151 - 1.18713i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (1.09 + 1.89i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.879 + 1.52i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.05 - 1.82i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.88 + 4.99i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.820 + 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.33 + 2.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.21 + 5.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 + (-2.56 + 4.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.14T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + (-5.89 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.72 - 2.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.59 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.87 - 6.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (7.75 - 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.561 + 0.971i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.96T + 83T^{2} \)
89 \( 1 + (-0.573 - 0.992i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.97T + 97T^{2} \)
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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

−10.32722345106143925351662677287, −9.370077566814129526873652076358, −8.648623607312374155165243981327, −7.85393730178682292883308460848, −6.74384502107590723426557680719, −5.93101955299640339810650154856, −3.98695217612706761310461563794, −2.80787146396598819215575541938, −2.22785786452515907565182451515, −0.853671758562385527432953524487, 1.62522407169955369505322153151, 3.69571411363732470667833999736, 4.73155189098100163473561871852, 5.60589964704065411291038783734, 6.69792323491332796414820081447, 7.51000306532162450512909826561, 8.530418744560861332241066980316, 9.192181146286901120380174530537, 9.723238677073559872716231636619, 10.22940627416415395709724418238

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