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Properties

Label 2-637-7.2-c1-0-0
Degree $2$
Conductor $637$
Sign $-0.900 + 0.435i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0711926 - 0.310360i\)
\(L(\frac12)\) \(\approx\) \(0.0711926 - 0.310360i\)
\(L(\frac{3}{2})\) 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

−11.20429472945710188611704796126, −9.696501337584584842996265322290, −9.416110157397763868608735682273, −7.998371302879599356972199507859, −7.33713628159515439582367540000, −6.84933324681113511741501205477, −6.25796121201837760286378354510, −5.07922395300277845556670999076, −3.59034144181585007869333205767, −1.63037130036487967972601267287, 0.24882245791168718362197934149, 1.66341684139127248045674183310, 3.47258557122457648877907316510, 4.02811741602913102855943171909, 5.24970949064032719792238086663, 6.30993165573031923717680619862, 8.052098556459864990468137641238, 8.710134413732009993839962336437, 9.291986945269575700771934954089, 10.37778765906698020037946425959

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