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Properties

Label 2-637-91.9-c1-0-31
Degree $2$
Conductor $637$
Sign $0.586 + 0.810i$
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  + (0.425 + 0.737i)2-s − 0.661·3-s + (0.637 − 1.10i)4-s + (1.72 − 2.98i)5-s + (−0.281 − 0.487i)6-s + 2.78·8-s − 2.56·9-s + 2.92·10-s − 0.897·11-s + (−0.421 + 0.730i)12-s + (3.07 + 1.88i)13-s + (−1.13 + 1.97i)15-s + (−0.0891 − 0.154i)16-s + (0.968 − 1.67i)17-s + (−1.09 − 1.88i)18-s − 1.03·19-s + ⋯
L(s)  = 1  + (0.300 + 0.521i)2-s − 0.381·3-s + (0.318 − 0.552i)4-s + (0.769 − 1.33i)5-s + (−0.114 − 0.198i)6-s + 0.985·8-s − 0.854·9-s + 0.926·10-s − 0.270·11-s + (−0.121 + 0.210i)12-s + (0.852 + 0.522i)13-s + (−0.293 + 0.508i)15-s + (−0.0222 − 0.0386i)16-s + (0.234 − 0.406i)17-s + (−0.257 − 0.445i)18-s − 0.238·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.586 + 0.810i$
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: \((2,\ 637,\ (\ :1/2),\ 0.586 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62341 - 0.828980i\)
\(L(\frac12)\) \(\approx\) \(1.62341 - 0.828980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.07 - 1.88i)T \)
good2 \( 1 + (-0.425 - 0.737i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 0.661T + 3T^{2} \)
5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.897T + 11T^{2} \)
17 \( 1 + (-0.968 + 1.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.917 + 1.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.56 + 7.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.66 - 4.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.95 - 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.59 + 6.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.69 - 8.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.255 - 0.442i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + (-1.72 - 2.98i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.45 - 9.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.253 - 0.438i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42504436448759828265039372960, −9.546990953311222396417085575394, −8.702914726793109289280420229966, −7.85622324256990656720396621369, −6.39698516701194081124618367648, −5.94768636525059477969266908178, −5.14521996825713427657431521611, −4.33324314291163581490899471342, −2.31287275318942083955025245895, −0.978092535452226539961957195733, 1.95351343460654125360533009685, 3.00723516432002903391718638565, 3.71742124689205621753508601015, 5.42653951482576987162561703769, 6.13282232531572047249617343071, 7.07659081994817375613212737150, 7.961449512985690023699772092801, 9.047609167323217875002776354675, 10.42679286645430724216047926805, 10.70316312995602718401276433805

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