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Properties

Label 2-637-91.19-c1-0-28
Degree $2$
Conductor $637$
Sign $0.931 - 0.362i$
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  + (1.33 + 0.357i)2-s + 1.07i·3-s + (−0.0814 − 0.0470i)4-s + (2.77 − 0.742i)5-s + (−0.383 + 1.42i)6-s + (−2.04 − 2.04i)8-s + 1.85·9-s + 3.96·10-s + (−1.00 − 1.00i)11-s + (0.0504 − 0.0873i)12-s + (3.54 + 0.634i)13-s + (0.796 + 2.97i)15-s + (−1.90 − 3.29i)16-s + (2.09 − 3.63i)17-s + (2.46 + 0.661i)18-s + (4.35 + 4.35i)19-s + ⋯
L(s)  = 1  + (0.942 + 0.252i)2-s + 0.618i·3-s + (−0.0407 − 0.0235i)4-s + (1.23 − 0.332i)5-s + (−0.156 + 0.583i)6-s + (−0.722 − 0.722i)8-s + 0.616·9-s + 1.25·10-s + (−0.304 − 0.304i)11-s + (0.0145 − 0.0252i)12-s + (0.984 + 0.176i)13-s + (0.205 + 0.767i)15-s + (−0.475 − 0.823i)16-s + (0.509 − 0.881i)17-s + (0.581 + 0.155i)18-s + (1.00 + 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.931 - 0.362i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70000 + 0.507271i\)
\(L(\frac12)\) \(\approx\) \(2.70000 + 0.507271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.54 - 0.634i)T \)
good2 \( 1 + (-1.33 - 0.357i)T + (1.73 + i)T^{2} \)
3 \( 1 - 1.07iT - 3T^{2} \)
5 \( 1 + (-2.77 + 0.742i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.00 + 1.00i)T + 11iT^{2} \)
17 \( 1 + (-2.09 + 3.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.35 - 4.35i)T + 19iT^{2} \)
23 \( 1 + (6.77 - 3.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.441 + 0.764i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.237 - 0.886i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.92 + 7.19i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (11.4 - 3.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.809 + 0.467i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.808 - 3.01i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.26 + 2.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.51 + 5.65i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 0.0854iT - 61T^{2} \)
67 \( 1 + (-0.728 + 0.728i)T - 67iT^{2} \)
71 \( 1 + (-2.79 - 0.750i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.75 + 0.737i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.71 - 8.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.54 - 1.54i)T + 83iT^{2} \)
89 \( 1 + (4.75 + 1.27i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.37 - 8.87i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.36164787085493034065832733145, −9.690050323955850652368964371812, −9.311117168160387213624415887422, −7.997503224772173954282079900345, −6.69474217099567413187540182569, −5.62293052426313227563466374986, −5.39066623178163008462808940529, −4.15835593918699339276392328286, −3.30871373926458340997452000231, −1.49913990275473719792939935108, 1.62292908750956356022541943131, 2.71015345340308109224486196244, 3.90378004620544701543402790341, 5.05785400567050321204476640733, 5.98426370438369672499796593130, 6.60136970416666682072755039011, 7.85378505858517092399366726984, 8.758959605722323013749717432839, 9.895615078362491822942806392047, 10.45939321519128693205865920593

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