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Properties

Label 2-637-91.81-c1-0-37
Degree $2$
Conductor $637$
Sign $-0.324 + 0.945i$
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  + (1.19 − 2.06i)2-s + 2.75·3-s + (−1.85 − 3.20i)4-s + (0.491 + 0.850i)5-s + (3.28 − 5.69i)6-s − 4.06·8-s + 4.57·9-s + 2.34·10-s − 0.587·11-s + (−5.09 − 8.82i)12-s + (−2.39 + 2.69i)13-s + (1.35 + 2.34i)15-s + (−1.15 + 1.99i)16-s + (−3.22 − 5.58i)17-s + (5.45 − 9.45i)18-s + 3.82·19-s + ⋯
L(s)  = 1  + (0.844 − 1.46i)2-s + 1.58·3-s + (−0.925 − 1.60i)4-s + (0.219 + 0.380i)5-s + (1.34 − 2.32i)6-s − 1.43·8-s + 1.52·9-s + 0.741·10-s − 0.177·11-s + (−1.47 − 2.54i)12-s + (−0.663 + 0.748i)13-s + (0.348 + 0.604i)15-s + (−0.288 + 0.498i)16-s + (−0.782 − 1.35i)17-s + (1.28 − 2.22i)18-s + 0.877·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.324 + 0.945i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.324 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09578 - 2.93478i\)
\(L(\frac12)\) \(\approx\) \(2.09578 - 2.93478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 2.69i)T \)
good2 \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.75T + 3T^{2} \)
5 \( 1 + (-0.491 - 0.850i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.587T + 11T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 + (4.13 - 7.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.98 - 3.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.49 - 2.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.877 - 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.83 - 3.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.19 - 5.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.17 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.212 - 0.368i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.00 - 5.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.20T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 + (1.80 - 3.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.46 + 4.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.39 + 2.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + (1.04 - 1.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.84 + 6.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.20191663217072024872166787372, −9.614134440101704674980559633898, −9.031690461089117871164551586103, −7.76961000084045241619079786000, −6.86992702818170139894133386171, −5.21928711606582431499280948284, −4.30880335296372158585092852009, −3.24029792458838389282794079176, −2.64586874865612167233941198747, −1.70018893531514244891975146749, 2.25400567755813143034188057719, 3.51532467660400654985259482081, 4.37625520513206671135161555368, 5.38743016778553271261975237130, 6.43670166354613625656057457561, 7.43105518863267110894059917507, 8.137436262557387223355342975252, 8.619271177109991079698889772786, 9.578263712540077930859947670022, 10.57254271953480297893811384964

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