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Properties

Label 2-637-91.81-c1-0-0
Degree $2$
Conductor $637$
Sign $-0.498 + 0.866i$
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  + (−0.777 + 1.34i)2-s + 0.489·3-s + (−0.208 − 0.361i)4-s + (−0.595 − 1.03i)5-s + (−0.380 + 0.658i)6-s − 2.46·8-s − 2.76·9-s + 1.85·10-s + 2.11·11-s + (−0.102 − 0.176i)12-s + (−2.86 + 2.19i)13-s + (−0.291 − 0.504i)15-s + (2.33 − 4.03i)16-s + (−0.453 − 0.784i)17-s + (2.14 − 3.71i)18-s − 6.69·19-s + ⋯
L(s)  = 1  + (−0.549 + 0.952i)2-s + 0.282·3-s + (−0.104 − 0.180i)4-s + (−0.266 − 0.461i)5-s + (−0.155 + 0.268i)6-s − 0.870·8-s − 0.920·9-s + 0.585·10-s + 0.638·11-s + (−0.0294 − 0.0510i)12-s + (−0.793 + 0.608i)13-s + (−0.0752 − 0.130i)15-s + (0.582 − 1.00i)16-s + (−0.109 − 0.190i)17-s + (0.505 − 0.876i)18-s − 1.53·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0449028 - 0.0776061i\)
\(L(\frac12)\) \(\approx\) \(0.0449028 - 0.0776061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.86 - 2.19i)T \)
good2 \( 1 + (0.777 - 1.34i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 0.489T + 3T^{2} \)
5 \( 1 + (0.595 + 1.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
17 \( 1 + (0.453 + 0.784i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.69T + 19T^{2} \)
23 \( 1 + (1.79 - 3.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.49 - 4.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.768 - 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.59 + 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.26T + 61T^{2} \)
67 \( 1 + 3.74T + 67T^{2} \)
71 \( 1 + (-1.26 + 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.86 - 4.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (8.87 - 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.10 + 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.30218448454239491172259865440, −9.891325448710144452771031272792, −9.089669685014393241013983138617, −8.483158929227223129414089670067, −7.81635348668651335919757922295, −6.77073695025526243949478430386, −6.08647924824265490367759968919, −4.87758087146219567160680310188, −3.66211319491247837514418765130, −2.27899805487584632284515828357, 0.05116712790025936450582635572, 1.96998288859616410485813180940, 2.91632199760739685895305683439, 3.89980429088985711781217479535, 5.47623157608475239128225843585, 6.43095297211112487469720873946, 7.50048171145342333349514489231, 8.685729755625635822057786598077, 9.075105865449830770888109327337, 10.21779175088751059960162603650

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