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Properties

Label 2-637-91.30-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.780 - 0.625i$
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  + (−0.156 + 0.0904i)2-s + 1.82·3-s + (−0.983 + 1.70i)4-s + (−2.32 − 1.34i)5-s + (−0.285 + 0.165i)6-s − 0.717i·8-s + 0.334·9-s + 0.485·10-s + 2.69i·11-s + (−1.79 + 3.11i)12-s + (−1.92 + 3.05i)13-s + (−4.24 − 2.45i)15-s + (−1.90 − 3.29i)16-s + (−2.38 + 4.12i)17-s + (−0.0523 + 0.0302i)18-s + 0.188i·19-s + ⋯
L(s)  = 1  + (−0.110 + 0.0639i)2-s + 1.05·3-s + (−0.491 + 0.851i)4-s + (−1.04 − 0.600i)5-s + (−0.116 + 0.0673i)6-s − 0.253i·8-s + 0.111·9-s + 0.153·10-s + 0.812i·11-s + (−0.518 + 0.898i)12-s + (−0.532 + 0.846i)13-s + (−1.09 − 0.633i)15-s + (−0.475 − 0.823i)16-s + (−0.577 + 1.00i)17-s + (−0.0123 + 0.00712i)18-s + 0.0432i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253916 + 0.722411i\)
\(L(\frac12)\) \(\approx\) \(0.253916 + 0.722411i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 - 3.05i)T \)
good2 \( 1 + (0.156 - 0.0904i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 1.82T + 3T^{2} \)
5 \( 1 + (2.32 + 1.34i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.69iT - 11T^{2} \)
17 \( 1 + (2.38 - 4.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 0.188iT - 19T^{2} \)
23 \( 1 + (-2.19 - 3.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.20 - 1.84i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.88 + 3.97i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.70 + 2.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.00 + 6.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.60 + 0.924i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.53 - 6.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.57 - 3.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.411T + 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 + (-2.89 + 1.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.3 + 7.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + (-5.10 + 2.94i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.390 - 0.225i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.02607104980464017004228666468, −9.582972726579782943546212850010, −8.984518960132669713267044531763, −8.388569628828146434436850809278, −7.61278928490473224192819145919, −6.98324590410424083493149290175, −5.09963409112086824667752272653, −4.08275285450281261781672591604, −3.55104586735937532320250424055, −2.10826383204110396035777892323, 0.36813195881527378916700734866, 2.48759101245439122364118352947, 3.37527833821535965690534311782, 4.49580087977057359092813563951, 5.63341095649794509102858937524, 6.82040199361926037929758239655, 7.903520118346946244730008481621, 8.410571910818371516814966395206, 9.368137482295905777996082574666, 10.06704074202246897608080362739

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