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Properties

Label 2-637-91.16-c1-0-29
Degree $2$
Conductor $637$
Sign $0.888 - 0.458i$
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  + 2.43·2-s + (−0.376 + 0.652i)3-s + 3.91·4-s + (−0.170 + 0.295i)5-s + (−0.916 + 1.58i)6-s + 4.65·8-s + (1.21 + 2.10i)9-s + (−0.415 + 0.719i)10-s + (1.21 − 2.10i)11-s + (−1.47 + 2.55i)12-s + (2.50 − 2.59i)13-s + (−0.128 − 0.222i)15-s + 3.49·16-s − 1.94·17-s + (2.95 + 5.12i)18-s + (3.14 + 5.44i)19-s + ⋯
L(s)  = 1  + 1.71·2-s + (−0.217 + 0.376i)3-s + 1.95·4-s + (−0.0763 + 0.132i)5-s + (−0.374 + 0.647i)6-s + 1.64·8-s + (0.405 + 0.702i)9-s + (−0.131 + 0.227i)10-s + (0.366 − 0.635i)11-s + (−0.425 + 0.737i)12-s + (0.693 − 0.720i)13-s + (−0.0332 − 0.0575i)15-s + 0.874·16-s − 0.472·17-s + (0.697 + 1.20i)18-s + (0.721 + 1.24i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.64728 + 0.884989i\)
\(L(\frac12)\) \(\approx\) \(3.64728 + 0.884989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 + 2.59i)T \)
good2 \( 1 - 2.43T + 2T^{2} \)
3 \( 1 + (0.376 - 0.652i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.170 - 0.295i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.21 + 2.10i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 + (-3.14 - 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 + (2.22 + 3.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.987 - 1.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.62T + 37T^{2} \)
41 \( 1 + (6.26 + 10.8i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.20 + 7.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.50 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.746 + 1.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.626T + 59T^{2} \)
61 \( 1 + (-0.571 - 0.990i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.79 + 4.84i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.74 - 8.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.95 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.23 - 3.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + (5.13 - 8.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.74833892856182825649139863565, −10.33761155564586333907837675783, −8.850394954106777423045605869432, −7.70838196413690120060974648499, −6.76823488227840940899961587427, −5.64776514438598218004393407531, −5.29531250006584164448681192751, −3.93284986339965082057327462709, −3.48674802619464543323935952763, −1.95099558375382084106402016846, 1.60351469940836083128015690356, 3.02203157493796540589639431562, 4.15392921758725349620203313955, 4.75756922848531031708232170027, 6.00517874651526705140915202405, 6.68002952262707555172482056629, 7.27963147305081769627625278650, 8.780743177160546556945942434453, 9.737637830554316423351957163948, 11.05038810128642534973445110496

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