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Properties

Label 2-637-91.76-c1-0-3
Degree $2$
Conductor $637$
Sign $0.782 - 0.623i$
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  + (0.638 − 2.38i)2-s + (0.146 − 0.0844i)3-s + (−3.53 − 2.04i)4-s + (−1.74 + 1.74i)5-s + (−0.107 − 0.402i)6-s + (−3.63 + 3.63i)8-s + (−1.48 + 2.57i)9-s + (3.04 + 5.26i)10-s + (−4.40 − 1.17i)11-s − 0.689·12-s + (1.54 + 3.25i)13-s + (−0.107 + 0.402i)15-s + (2.24 + 3.89i)16-s + (−0.0563 + 0.0976i)17-s + (5.18 + 5.18i)18-s + (0.891 + 3.32i)19-s + ⋯
L(s)  = 1  + (0.451 − 1.68i)2-s + (0.0844 − 0.0487i)3-s + (−1.76 − 1.02i)4-s + (−0.780 + 0.780i)5-s + (−0.0440 − 0.164i)6-s + (−1.28 + 1.28i)8-s + (−0.495 + 0.857i)9-s + (0.962 + 1.66i)10-s + (−1.32 − 0.355i)11-s − 0.198·12-s + (0.429 + 0.903i)13-s + (−0.0278 + 0.103i)15-s + (0.562 + 0.973i)16-s + (−0.0136 + 0.0236i)17-s + (1.22 + 1.22i)18-s + (0.204 + 0.763i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.782 - 0.623i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.782 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454099 + 0.158745i\)
\(L(\frac12)\) \(\approx\) \(0.454099 + 0.158745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 - 3.25i)T \)
good2 \( 1 + (-0.638 + 2.38i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-0.146 + 0.0844i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.74 - 1.74i)T - 5iT^{2} \)
11 \( 1 + (4.40 + 1.17i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0563 - 0.0976i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.891 - 3.32i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.565 + 0.326i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.82 + 4.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.91 - 3.91i)T - 31iT^{2} \)
37 \( 1 + (1.50 + 0.402i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (10.6 + 2.86i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.08 - 3.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.18 - 4.18i)T + 47iT^{2} \)
53 \( 1 - 4.82T + 53T^{2} \)
59 \( 1 + (3.81 - 1.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (13.2 + 7.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 + 6.07i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.56 + 0.955i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.77 + 1.77i)T + 73iT^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + (3.34 - 3.34i)T - 83iT^{2} \)
89 \( 1 + (2.24 - 8.39i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.04 - 3.89i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.83803368871992554869700042067, −10.34855215912204053456408310517, −9.199977428845685826352671212171, −8.179968209077925697913918454836, −7.33667585773888861383486286092, −5.79078773779472295867833944570, −4.80803042463095619666938948993, −3.70597488372236977803997732155, −2.93856799032654799212465404878, −1.91405660693858360505515787742, 0.21617698494488081208209078517, 3.18681031584030718380635517367, 4.27493747200799944303743214538, 5.22712703974702075659181798501, 5.80457035521777971883670897577, 7.07413285351748637535207680043, 7.70549511426306719851055639188, 8.551186239324766706669316119957, 9.006404223556255380083578104891, 10.34261900958109480404527040413

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