Properties

Label 2-637-13.12-c1-0-39
Degree $2$
Conductor $637$
Sign $-0.893 + 0.449i$
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  − 2.28i·2-s + 3.15·3-s − 3.22·4-s − 2.12i·5-s − 7.19i·6-s + 2.78i·8-s + 6.92·9-s − 4.85·10-s + 0.308i·11-s − 10.1·12-s + (−3.22 + 1.62i)13-s − 6.69i·15-s − 0.0699·16-s + 1.77·17-s − 15.8i·18-s − 1.78i·19-s + ⋯
L(s)  = 1  − 1.61i·2-s + 1.81·3-s − 1.61·4-s − 0.950i·5-s − 2.93i·6-s + 0.985i·8-s + 2.30·9-s − 1.53·10-s + 0.0930i·11-s − 2.92·12-s + (−0.893 + 0.449i)13-s − 1.72i·15-s − 0.0174·16-s + 0.430·17-s − 3.72i·18-s − 0.408i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575692 - 2.42348i\)
\(L(\frac12)\) \(\approx\) \(0.575692 - 2.42348i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.22 - 1.62i)T \)
good2 \( 1 + 2.28iT - 2T^{2} \)
3 \( 1 - 3.15T + 3T^{2} \)
5 \( 1 + 2.12iT - 5T^{2} \)
11 \( 1 - 0.308iT - 11T^{2} \)
17 \( 1 - 1.77T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 - 4.60iT - 31T^{2} \)
37 \( 1 + 5.54iT - 37T^{2} \)
41 \( 1 - 6.72iT - 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 - 9.51iT - 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + 8.12iT - 59T^{2} \)
61 \( 1 + 3.44T + 61T^{2} \)
67 \( 1 + 12.6iT - 67T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + 7.92T + 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + 1.65iT - 89T^{2} \)
97 \( 1 - 7.66iT - 97T^{2} \)
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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

−9.908651673365704613780451456558, −9.475229401147307032910031207339, −8.808074540104131495022894151468, −8.074613956867905859326221996747, −7.00338551540200121550635657216, −4.89773961399094180224063342845, −4.20014759570961887573140485826, −3.16043400913848469774021991788, −2.31665331123374573648386434864, −1.27911402945754400666877495035, 2.35348291911943812986449471216, 3.36090369820137091405352810323, 4.47775075558411272574369073383, 5.76283749226617412902389167231, 6.98416164321856509338315817346, 7.37124463144357551405640885234, 8.171906475501774756357134318209, 8.796681742684865703150866708805, 9.791348161882147160663566664706, 10.38322020037440224643635544923

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