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Properties

Label 2-637-91.9-c1-0-22
Degree $2$
Conductor $637$
Sign $0.471 + 0.882i$
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  + (0.289 + 0.502i)2-s − 1.89·3-s + (0.831 − 1.44i)4-s + (−0.736 + 1.27i)5-s + (−0.548 − 0.950i)6-s + 2.12·8-s + 0.579·9-s − 0.854·10-s + 0.579·11-s + (−1.57 + 2.72i)12-s + (−0.128 − 3.60i)13-s + (1.39 − 2.41i)15-s + (−1.04 − 1.81i)16-s + (0.598 − 1.03i)17-s + (0.168 + 0.291i)18-s − 0.460·19-s + ⋯
L(s)  = 1  + (0.204 + 0.355i)2-s − 1.09·3-s + (0.415 − 0.720i)4-s + (−0.329 + 0.570i)5-s + (−0.223 − 0.387i)6-s + 0.751·8-s + 0.193·9-s − 0.270·10-s + 0.174·11-s + (−0.454 + 0.787i)12-s + (−0.0357 − 0.999i)13-s + (0.359 − 0.623i)15-s + (−0.261 − 0.453i)16-s + (0.145 − 0.251i)17-s + (0.0396 + 0.0686i)18-s − 0.105·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.471 + 0.882i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.471 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861933 - 0.516829i\)
\(L(\frac12)\) \(\approx\) \(0.861933 - 0.516829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.128 + 3.60i)T \)
good2 \( 1 + (-0.289 - 0.502i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 1.89T + 3T^{2} \)
5 \( 1 + (0.736 - 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.579T + 11T^{2} \)
17 \( 1 + (-0.598 + 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 0.460T + 19T^{2} \)
23 \( 1 + (1.18 + 2.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.44 + 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.22 + 3.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.58 + 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.02 + 6.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.75 + 9.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.69 - 8.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.120 - 0.208i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.72T + 61T^{2} \)
67 \( 1 + 1.44T + 67T^{2} \)
71 \( 1 + (-6.25 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.84 - 3.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.03 - 13.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (1.24 + 2.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.82 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54087804395691024178131794045, −9.998229171609025365026002359339, −8.593327061210134546466172361920, −7.36615536854117370901645667668, −6.78864333083529584085502565648, −5.71893162263153022911162748028, −5.39063775777190702686948514594, −4.03582847618678352470385246225, −2.48272078879581929991038210113, −0.61498371685413355659004989304, 1.47003209809559148518823047527, 3.09820664560941021363336372774, 4.32324228638797322867602781170, 5.03485345398143778064823912804, 6.33010015300861960803316643531, 6.98246751193299329179854128527, 8.153358642086044038870938033041, 8.880668617204004762065250086422, 10.17147538091222620459639221431, 11.03984552273993751628501485457

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