Properties

Label 2-637-13.4-c1-0-5
Degree $2$
Conductor $637$
Sign $0.993 - 0.114i$
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  + (−1.99 − 1.15i)2-s + (0.736 − 1.27i)3-s + (1.65 + 2.86i)4-s + 0.847i·5-s + (−2.93 + 1.69i)6-s − 3.00i·8-s + (0.414 + 0.718i)9-s + (0.975 − 1.69i)10-s + (−1.30 − 0.751i)11-s + 4.86·12-s + (−2.92 + 2.11i)13-s + (1.08 + 0.624i)15-s + (−0.156 + 0.271i)16-s + (1.03 + 1.79i)17-s − 1.90i·18-s + (−0.0410 + 0.0237i)19-s + ⋯
L(s)  = 1  + (−1.41 − 0.814i)2-s + (0.425 − 0.736i)3-s + (0.826 + 1.43i)4-s + 0.378i·5-s + (−1.19 + 0.692i)6-s − 1.06i·8-s + (0.138 + 0.239i)9-s + (0.308 − 0.534i)10-s + (−0.392 − 0.226i)11-s + 1.40·12-s + (−0.810 + 0.585i)13-s + (0.279 + 0.161i)15-s + (−0.0391 + 0.0678i)16-s + (0.251 + 0.435i)17-s − 0.450i·18-s + (−0.00942 + 0.00544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674464 + 0.0386717i\)
\(L(\frac12)\) \(\approx\) \(0.674464 + 0.0386717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.92 - 2.11i)T \)
good2 \( 1 + (1.99 + 1.15i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.736 + 1.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.847iT - 5T^{2} \)
11 \( 1 + (1.30 + 0.751i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.03 - 1.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0410 - 0.0237i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.90 - 6.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.679 - 1.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.86iT - 31T^{2} \)
37 \( 1 + (-5.80 - 3.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.67 + 5.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.360iT - 47T^{2} \)
53 \( 1 - 2.71T + 53T^{2} \)
59 \( 1 + (1.42 - 0.820i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.26 + 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.76 - 1.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.3 + 7.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (15.1 + 8.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.369 + 0.213i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.46615841326142916564432836745, −9.784118855874632330986104325249, −8.895459167878458909477326516820, −8.023807091250307358874510471887, −7.48950900995470973278828613104, −6.62894084884670103138301794489, −5.06178144367208174832199003962, −3.33516194764847129033545780921, −2.34225724933901819590696574409, −1.40324194851048462398560765864, 0.58508244378662324678390969674, 2.51122752624436394999967206161, 4.09721959558265029418632810045, 5.19911209617390517976929343764, 6.34917621722010618351403051839, 7.31706148150128774477705415628, 8.122840559320509823331693613342, 8.808943071086806150317399505252, 9.696306584574302706158761055920, 10.02719243232972149996955297999

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