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Properties

Label 2-637-13.12-c1-0-2
Degree $2$
Conductor $637$
Sign $-0.409 - 0.912i$
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  − 0.680i·2-s − 2.33·3-s + 1.53·4-s + 3.24i·5-s + 1.58i·6-s − 2.40i·8-s + 2.44·9-s + 2.20·10-s + 5.33i·11-s − 3.58·12-s + (−1.47 − 3.28i)13-s − 7.57i·15-s + 1.43·16-s − 2.66·17-s − 1.66i·18-s + 0.696i·19-s + ⋯
L(s)  = 1  − 0.480i·2-s − 1.34·3-s + 0.768·4-s + 1.45i·5-s + 0.648i·6-s − 0.850i·8-s + 0.816·9-s + 0.698·10-s + 1.60i·11-s − 1.03·12-s + (−0.409 − 0.912i)13-s − 1.95i·15-s + 0.359·16-s − 0.647·17-s − 0.392i·18-s + 0.159i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.409 - 0.912i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.409 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350358 + 0.541441i\)
\(L(\frac12)\) \(\approx\) \(0.350358 + 0.541441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 + 3.28i)T \)
good2 \( 1 + 0.680iT - 2T^{2} \)
3 \( 1 + 2.33T + 3T^{2} \)
5 \( 1 - 3.24iT - 5T^{2} \)
11 \( 1 - 5.33iT - 11T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
19 \( 1 - 0.696iT - 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 - 5.45iT - 31T^{2} \)
37 \( 1 - 6.69iT - 37T^{2} \)
41 \( 1 - 2.21iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 - 5.79iT - 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 + 2.98iT - 59T^{2} \)
61 \( 1 - 2.19T + 61T^{2} \)
67 \( 1 - 8.70iT - 67T^{2} \)
71 \( 1 - 8.56iT - 71T^{2} \)
73 \( 1 + 5.88iT - 73T^{2} \)
79 \( 1 - 0.910T + 79T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + 0.986iT - 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−10.94714006056364928099733340114, −10.13063115487862053660067507769, −9.925393658345950297973616830686, −7.82358186534123633896667269830, −7.03013269230656290014711796557, −6.48307811957904718097376739548, −5.61957985781280014134908178811, −4.31557957178209205684626581398, −2.96923133409905559320914524917, −1.90224961669680199924580662220, 0.38189717922567172418118627188, 1.97445200829911135925852425046, 4.02317669371540199963807253575, 5.13504507294540007868483836310, 5.85184547182635455763681387405, 6.35416903443135751154514583306, 7.58469545802668704392312096138, 8.491889568320255682817615180679, 9.309622601100474632387668787212, 10.59590584916370657148774393103

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