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Properties

Label 2-637-13.3-c1-0-18
Degree $2$
Conductor $637$
Sign $0.973 + 0.227i$
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  + (0.425 + 0.737i)2-s + (0.330 + 0.572i)3-s + (0.637 − 1.10i)4-s − 3.44·5-s + (−0.281 + 0.487i)6-s + 2.78·8-s + (1.28 − 2.21i)9-s + (−1.46 − 2.53i)10-s + (0.448 + 0.777i)11-s + 0.843·12-s + (3.07 − 1.88i)13-s + (−1.13 − 1.97i)15-s + (−0.0891 − 0.154i)16-s + (0.968 − 1.67i)17-s + 2.18·18-s + (0.519 − 0.898i)19-s + ⋯
L(s)  = 1  + (0.300 + 0.521i)2-s + (0.190 + 0.330i)3-s + (0.318 − 0.552i)4-s − 1.53·5-s + (−0.114 + 0.198i)6-s + 0.985·8-s + (0.427 − 0.739i)9-s + (−0.463 − 0.802i)10-s + (0.135 + 0.234i)11-s + 0.243·12-s + (0.852 − 0.522i)13-s + (−0.293 − 0.508i)15-s + (−0.0222 − 0.0386i)16-s + (0.234 − 0.406i)17-s + 0.514·18-s + (0.119 − 0.206i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.973 + 0.227i$
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: \((2,\ 637,\ (\ :1/2),\ 0.973 + 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69204 - 0.195193i\)
\(L(\frac12)\) \(\approx\) \(1.69204 - 0.195193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.07 + 1.88i)T \)
good2 \( 1 + (-0.425 - 0.737i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.330 - 0.572i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 + (-0.448 - 0.777i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.968 + 1.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.519 + 0.898i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.917 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.13T + 31T^{2} \)
37 \( 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.66 + 4.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.95 + 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 + (0.255 - 0.442i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.718 + 1.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.22 - 7.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.72 + 2.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.253 + 0.438i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.50802584959899506452063781085, −9.826133593007085327221591238358, −8.554094582872295498404525104265, −7.88147879205476349757075308948, −6.89060962199816916086229275611, −6.23824050031603354714177891347, −4.82273011956770474829396825167, −4.15836346654335059506933594316, −3.07634512198213374880369624647, −0.949522021807354355401760335051, 1.54195222339506745114845247166, 3.01331195750254048415041859487, 3.92912413072989850208013847787, 4.58204965723302082594607634166, 6.28363043479190534648720001033, 7.38445748707813653936363781759, 7.932873295874432540146230573045, 8.477775860076572967741699863455, 9.957284493029775115233486073852, 11.07094532380806159396062954222

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