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Properties

Label 2-637-13.9-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.997 - 0.0662i$
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  + (1.19 − 2.06i)2-s + (−1.37 + 2.38i)3-s + (−1.85 − 3.20i)4-s − 0.982·5-s + (3.28 + 5.69i)6-s − 4.06·8-s + (−2.28 − 3.95i)9-s + (−1.17 + 2.03i)10-s + (0.293 − 0.509i)11-s + 10.1·12-s + (−2.39 − 2.69i)13-s + (1.35 − 2.34i)15-s + (−1.15 + 1.99i)16-s + (−3.22 − 5.58i)17-s − 10.9·18-s + (−1.91 − 3.31i)19-s + ⋯
L(s)  = 1  + (0.844 − 1.46i)2-s + (−0.794 + 1.37i)3-s + (−0.925 − 1.60i)4-s − 0.439·5-s + (1.34 + 2.32i)6-s − 1.43·8-s + (−0.761 − 1.31i)9-s + (−0.370 + 0.642i)10-s + (0.0886 − 0.153i)11-s + 2.94·12-s + (−0.663 − 0.748i)13-s + (0.348 − 0.604i)15-s + (−0.288 + 0.498i)16-s + (−0.782 − 1.35i)17-s − 2.57·18-s + (−0.438 − 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.997 - 0.0662i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.997 - 0.0662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0237770 + 0.716696i\)
\(L(\frac12)\) \(\approx\) \(0.0237770 + 0.716696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 + 2.69i)T \)
good2 \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.37 - 2.38i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.982T + 5T^{2} \)
11 \( 1 + (-0.293 + 0.509i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 + 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.13 - 7.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.98 + 3.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 + (0.877 - 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.83 + 3.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.19 + 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.34T + 47T^{2} \)
53 \( 1 - 0.425T + 53T^{2} \)
59 \( 1 + (-3.00 - 5.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.10 - 1.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.50 - 6.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.80 + 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.93T + 73T^{2} \)
79 \( 1 - 2.78T + 79T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + (1.04 - 1.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.84 - 6.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27255304313379267991831627645, −9.845584340106662003681702256961, −8.979458381950145459110378478650, −7.43361678762777666717330864909, −5.86489852662051966639061862651, −5.08094315472615074688098133796, −4.39109624051100852605796028385, −3.60963236269781726953614217277, −2.50399705313134471980680001288, −0.31277695431341298370700645824, 1.97773268463008196544616204740, 4.00601702728742262940727639712, 4.80668824796761430156960131097, 6.18561908721863658633074669499, 6.29574932248340982513244259816, 7.22900327588305698938840646427, 7.978191722451259856812790759156, 8.612414651379199301256497090486, 10.28768314508220989530270103726, 11.41802199117222291063904971497

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