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Properties

Label 2-637-13.4-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.470 + 0.882i$
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  + (0.713 + 0.411i)2-s + (1.33 − 2.30i)3-s + (−0.660 − 1.14i)4-s − 3.16i·5-s + (1.89 − 1.09i)6-s − 2.73i·8-s + (−2.03 − 3.53i)9-s + (1.30 − 2.25i)10-s + (5.14 + 2.97i)11-s − 3.51·12-s + (0.0766 + 3.60i)13-s + (−7.28 − 4.20i)15-s + (−0.195 + 0.338i)16-s + (1.34 + 2.33i)17-s − 3.35i·18-s + (−1.69 + 0.978i)19-s + ⋯
L(s)  = 1  + (0.504 + 0.291i)2-s + (0.767 − 1.33i)3-s + (−0.330 − 0.572i)4-s − 1.41i·5-s + (0.774 − 0.447i)6-s − 0.967i·8-s + (−0.679 − 1.17i)9-s + (0.411 − 0.713i)10-s + (1.55 + 0.895i)11-s − 1.01·12-s + (0.0212 + 0.999i)13-s + (−1.88 − 1.08i)15-s + (−0.0488 + 0.0845i)16-s + (0.327 + 0.567i)17-s − 0.791i·18-s + (−0.388 + 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21302 - 2.02071i\)
\(L(\frac12)\) \(\approx\) \(1.21302 - 2.02071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0766 - 3.60i)T \)
good2 \( 1 + (-0.713 - 0.411i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.33 + 2.30i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.16iT - 5T^{2} \)
11 \( 1 + (-5.14 - 2.97i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.34 - 2.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.69 - 0.978i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.36 - 2.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.15iT - 31T^{2} \)
37 \( 1 + (5.63 + 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.23 - 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.49 - 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.456iT - 47T^{2} \)
53 \( 1 - 0.399T + 53T^{2} \)
59 \( 1 + (4.16 - 2.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.578 + 1.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 + 3.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.90 + 2.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.30iT - 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 - 6.19iT - 83T^{2} \)
89 \( 1 + (3.08 + 1.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.96 + 1.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.889464533725839542082821738105, −9.176676173507499018115299751008, −8.642603984119553725187870866746, −7.57875405230273626737159111777, −6.63490774387456949712878987896, −5.95677992962297177374468927527, −4.59147805801929084173600291245, −3.96097988477808461339013931132, −1.82968650787409483783733539240, −1.18412687789622841180837581034, 2.72124699406750702728399954691, 3.34360490815164177812803210099, 3.91048297176101614049532282401, 5.03875570242014514782596729692, 6.30057939191136074119568251441, 7.43128295905528742861073881571, 8.586227209412960037351730695204, 9.028490011773746151166781141753, 10.18853671372564572455027227089, 10.74816466623257888513825611505

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