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Properties

Label 2-637-13.3-c1-0-17
Degree $2$
Conductor $637$
Sign $0.617 - 0.786i$
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  + (1.16 + 2.01i)2-s + (−1.15 − 1.99i)3-s + (−1.71 + 2.97i)4-s + 3.37·5-s + (2.69 − 4.66i)6-s − 3.34·8-s + (−1.16 + 2.01i)9-s + (3.92 + 6.80i)10-s + (−1.16 − 2.01i)11-s + 7.93·12-s + (0.408 + 3.58i)13-s + (−3.89 − 6.74i)15-s + (−0.466 − 0.808i)16-s + (2.72 − 4.72i)17-s − 5.43·18-s + (3.58 − 6.20i)19-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)2-s + (−0.666 − 1.15i)3-s + (−0.858 + 1.48i)4-s + 1.50·5-s + (1.09 − 1.90i)6-s − 1.18·8-s + (−0.388 + 0.673i)9-s + (1.24 + 2.15i)10-s + (−0.351 − 0.608i)11-s + 2.29·12-s + (0.113 + 0.993i)13-s + (−1.00 − 1.74i)15-s + (−0.116 − 0.202i)16-s + (0.661 − 1.14i)17-s − 1.28·18-s + (0.822 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09620 + 1.01933i\)
\(L(\frac12)\) \(\approx\) \(2.09620 + 1.01933i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.408 - 3.58i)T \)
good2 \( 1 + (-1.16 - 2.01i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.15 + 1.99i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.37T + 5T^{2} \)
11 \( 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.72 + 4.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.58 + 6.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.22 - 5.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.22 - 7.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + (1.52 + 2.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.468 - 0.812i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.04 + 3.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + (3.62 - 6.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.19 - 5.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.30 + 3.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.79 + 6.57i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.06T + 73T^{2} \)
79 \( 1 + 7.58T + 79T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 + (6.57 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.77 + 3.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92417432846741936171025278395, −9.507796158983861149597414380219, −8.825592048779301317447816808552, −7.29640587639758682251738853503, −7.10417673830165012319683845128, −6.17443488899378889804159123925, −5.49150699438762978378239730757, −4.94511416817988465295532594209, −3.03371630124749315740872860735, −1.39684944977077239307035708651, 1.47572498512007748167774136262, 2.71465563260493320135431246918, 3.82108781491715737917609565054, 4.85767353539839806071901066988, 5.52466898490866648135221072432, 6.12933594392751428547040942354, 8.042721997994948108429505932805, 9.569227284825654373202355974365, 10.02881473203841458105950706452, 10.39193177549276948630690102050

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