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Properties

Label 2-637-91.41-c1-0-30
Degree $2$
Conductor $637$
Sign $-0.426 + 0.904i$
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  + (−1.75 − 0.470i)2-s + (2.65 − 1.53i)3-s + (1.12 + 0.650i)4-s + (−0.787 − 0.787i)5-s + (−5.38 + 1.44i)6-s + (0.899 + 0.899i)8-s + (3.21 − 5.56i)9-s + (1.01 + 1.75i)10-s + (−0.0332 + 0.124i)11-s + 3.99·12-s + (3.59 + 0.291i)13-s + (−3.30 − 0.884i)15-s + (−2.45 − 4.25i)16-s + (0.136 − 0.236i)17-s + (−8.25 + 8.25i)18-s + (3.83 − 1.02i)19-s + ⋯
L(s)  = 1  + (−1.24 − 0.332i)2-s + (1.53 − 0.886i)3-s + (0.562 + 0.325i)4-s + (−0.352 − 0.352i)5-s + (−2.19 + 0.589i)6-s + (0.317 + 0.317i)8-s + (1.07 − 1.85i)9-s + (0.319 + 0.553i)10-s + (−0.0100 + 0.0374i)11-s + 1.15·12-s + (0.996 + 0.0807i)13-s + (−0.852 − 0.228i)15-s + (−0.613 − 1.06i)16-s + (0.0331 − 0.0574i)17-s + (−1.94 + 1.94i)18-s + (0.879 − 0.235i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665452 - 1.04895i\)
\(L(\frac12)\) \(\approx\) \(0.665452 - 1.04895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 - 0.291i)T \)
good2 \( 1 + (1.75 + 0.470i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-2.65 + 1.53i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.787 + 0.787i)T + 5iT^{2} \)
11 \( 1 + (0.0332 - 0.124i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.136 + 0.236i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.83 + 1.02i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.426 + 0.245i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.62 + 8.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.06 + 5.06i)T + 31iT^{2} \)
37 \( 1 + (1.03 - 3.85i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.31 - 8.63i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.44 - 4.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.14 - 4.14i)T - 47iT^{2} \)
53 \( 1 + 0.934T + 53T^{2} \)
59 \( 1 + (-1.78 - 6.64i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.18 - 0.316i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.793 + 2.96i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.0869 - 0.0869i)T - 73iT^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (8.07 + 8.07i)T + 83iT^{2} \)
89 \( 1 + (4.09 + 1.09i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.39 + 0.641i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.800943392462631157914043767479, −9.305933505957562707687025990681, −8.541364838947108709293982364518, −7.903499211567862451703917410909, −7.43148886489515994312338888918, −6.15923477952182227691823121272, −4.37504379863016820144332717320, −3.14072916582195973224482307349, −2.01220378527274403904902010704, −0.940131235382146361495093009811, 1.71085026299115817513173720364, 3.36460722395982678678810977988, 3.84478776723824672311426418615, 5.34085810111746898648656716889, 7.09071583373042894898044073514, 7.57405636546954136971378788606, 8.565623100522900343206812313862, 8.974437435260955341313714906739, 9.645624579095874455044073379785, 10.61697405394958150035760300579

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