Properties

Label 2-637-13.3-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.992 - 0.124i$
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  + (−0.777 − 1.34i)2-s + (−0.244 − 0.423i)3-s + (−0.208 + 0.361i)4-s + 1.19·5-s + (−0.380 + 0.658i)6-s − 2.46·8-s + (1.38 − 2.39i)9-s + (−0.926 − 1.60i)10-s + (−1.05 − 1.83i)11-s + 0.204·12-s + (−2.86 + 2.19i)13-s + (−0.291 − 0.504i)15-s + (2.33 + 4.03i)16-s + (−0.453 + 0.784i)17-s − 4.29·18-s + (3.34 − 5.79i)19-s + ⋯
L(s)  = 1  + (−0.549 − 0.952i)2-s + (−0.141 − 0.244i)3-s + (−0.104 + 0.180i)4-s + 0.532·5-s + (−0.155 + 0.268i)6-s − 0.870·8-s + (0.460 − 0.796i)9-s + (−0.292 − 0.507i)10-s + (−0.319 − 0.552i)11-s + 0.0589·12-s + (−0.793 + 0.608i)13-s + (−0.0752 − 0.130i)15-s + (0.582 + 1.00i)16-s + (−0.109 + 0.190i)17-s − 1.01·18-s + (0.767 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.992 - 0.124i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0536627 + 0.858331i\)
\(L(\frac12)\) \(\approx\) \(0.0536627 + 0.858331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.86 - 2.19i)T \)
good2 \( 1 + (0.777 + 1.34i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.244 + 0.423i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.19T + 5T^{2} \)
11 \( 1 + (1.05 + 1.83i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.453 - 0.784i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.34 + 5.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.79 + 3.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.28T + 31T^{2} \)
37 \( 1 + (2.49 + 4.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.768 - 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.18T + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.13 - 7.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.26 + 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.73T + 73T^{2} \)
79 \( 1 - 6.07T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (8.87 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.10 + 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.966453144001921960105630162507, −9.559499328061491349280843080954, −8.779058962112711746349886723130, −7.51893797151834570761818083622, −6.46749395699977994555413031890, −5.73980239168709505360386344520, −4.33099235934885801851925089884, −2.94296140772031489149230943743, −1.95517915759830916865595483719, −0.54480117693191687495316566520, 1.98402621044789217797760315432, 3.43087224757493627613410966375, 5.03529332865910572018579631645, 5.61093033995761390032615976983, 6.75861900837385444492159926521, 7.68226704327587728973834532054, 8.040392581417516686105459450090, 9.444301707306246521386676732635, 9.868858440389529446112814227094, 10.70691804507215496645663518914

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