Properties

Label 2-637-91.74-c1-0-1
Degree $2$
Conductor $637$
Sign $0.0575 - 0.998i$
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  − 2.74·2-s + (−0.682 − 1.18i)3-s + 5.51·4-s + (−0.370 − 0.641i)5-s + (1.87 + 3.23i)6-s − 9.63·8-s + (0.568 − 0.984i)9-s + (1.01 + 1.75i)10-s + (0.682 + 1.18i)11-s + (−3.76 − 6.51i)12-s + (0.301 + 3.59i)13-s + (−0.505 + 0.875i)15-s + 15.3·16-s − 4.14·17-s + (−1.55 + 2.69i)18-s + (−3.63 + 6.29i)19-s + ⋯
L(s)  = 1  − 1.93·2-s + (−0.393 − 0.682i)3-s + 2.75·4-s + (−0.165 − 0.287i)5-s + (0.763 + 1.32i)6-s − 3.40·8-s + (0.189 − 0.328i)9-s + (0.321 + 0.556i)10-s + (0.205 + 0.356i)11-s + (−1.08 − 1.88i)12-s + (0.0837 + 0.996i)13-s + (−0.130 + 0.226i)15-s + 3.84·16-s − 1.00·17-s + (−0.367 + 0.636i)18-s + (−0.833 + 1.44i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0575 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0575 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0575 - 0.998i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0575 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156304 + 0.147546i\)
\(L(\frac12)\) \(\approx\) \(0.156304 + 0.147546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.301 - 3.59i)T \)
good2 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 + (0.682 + 1.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.370 + 0.641i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.38 + 2.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + (0.627 - 1.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.92 - 5.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.28 - 3.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + (3.26 - 5.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.87 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.03 - 5.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 1.76T + 89T^{2} \)
97 \( 1 + (4.76 + 8.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.61093791118923783648366102920, −9.791735803128591836154009341356, −9.000171211902498752006739304994, −8.288274946915704553626773549814, −7.40540709921642098573204160479, −6.58078534147424720324816026727, −6.10585348998826593581116284136, −4.09542559838221610750002603225, −2.23057552458400994297437591355, −1.31389159898318514484699604796, 0.23670754907169003333456472033, 2.04551074346973380959626175496, 3.31271096930131691178799743818, 4.99409751990236933387070868652, 6.26837385517952032204545951631, 7.04688237108837418806253634072, 7.955398155893523658604431535533, 8.808918835116419083420014125592, 9.443063469861906985613485885236, 10.62601786688380032238232963445

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