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Properties

Label 2-637-7.4-c1-0-38
Degree $2$
Conductor $637$
Sign $0.386 - 0.922i$
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  + (−1.10 − 1.91i)2-s + (1.23 − 2.14i)3-s + (−1.44 + 2.51i)4-s + (−1.06 − 1.83i)5-s − 5.47·6-s + 1.98·8-s + (−1.56 − 2.70i)9-s + (−2.34 + 4.06i)10-s + (−2.39 + 4.14i)11-s + (3.58 + 6.21i)12-s − 13-s − 5.25·15-s + (0.697 + 1.20i)16-s + (−1.88 + 3.27i)17-s + (−3.45 + 5.98i)18-s + (−1.78 − 3.08i)19-s + ⋯
L(s)  = 1  + (−0.782 − 1.35i)2-s + (0.714 − 1.23i)3-s + (−0.724 + 1.25i)4-s + (−0.474 − 0.822i)5-s − 2.23·6-s + 0.703·8-s + (−0.520 − 0.901i)9-s + (−0.742 + 1.28i)10-s + (−0.721 + 1.25i)11-s + (1.03 + 1.79i)12-s − 0.277·13-s − 1.35·15-s + (0.174 + 0.301i)16-s + (−0.458 + 0.793i)17-s + (−0.814 + 1.41i)18-s + (−0.409 − 0.708i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367994 + 0.244783i\)
\(L(\frac12)\) \(\approx\) \(0.367994 + 0.244783i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (1.10 + 1.91i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.23 + 2.14i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.06 + 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.39 - 4.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.88 - 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 + 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (1.88 - 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.60 - 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + (-3.85 + 6.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.58 - 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.40T + 97T^{2} \)
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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

−9.919922467113062221922776997097, −8.800354276732977025488071893893, −8.456124833636656614242142079755, −7.59086993502241630895377042209, −6.69842323198391815053758237848, −4.94786583144003691102395949110, −3.75313566394283012263812277780, −2.33110500640930476114746507996, −1.82330263511639808724171843942, −0.26997157794483499771257263499, 2.90293569261585238618466839908, 3.75230987967463537343016969537, 5.11028615211351340958752621347, 5.99899572158627130997233637636, 7.14125369192307231609143865047, 7.88003329474859377838926078968, 8.610395059913690472479325606381, 9.345781470475511552999918643003, 10.12055222714213318614032846841, 10.85041753516643497059480385657

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