Properties

Label 2-637-49.43-c1-0-12
Degree $2$
Conductor $637$
Sign $-0.544 - 0.838i$
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  + (−0.118 + 0.149i)2-s + (1.11 + 0.539i)3-s + (0.436 + 1.91i)4-s + (0.156 + 0.0751i)5-s + (−0.213 + 0.102i)6-s + (0.0412 + 2.64i)7-s + (−0.680 − 0.327i)8-s + (−0.907 − 1.13i)9-s + (−0.0297 + 0.0143i)10-s + (−2.91 + 3.65i)11-s + (−0.542 + 2.37i)12-s + (0.623 − 0.781i)13-s + (−0.399 − 0.308i)14-s + (0.134 + 0.168i)15-s + (−3.40 + 1.64i)16-s + (−0.0503 + 0.220i)17-s + ⋯
L(s)  = 1  + (−0.0840 + 0.105i)2-s + (0.646 + 0.311i)3-s + (0.218 + 0.957i)4-s + (0.0697 + 0.0335i)5-s + (−0.0871 + 0.0419i)6-s + (0.0156 + 0.999i)7-s + (−0.240 − 0.115i)8-s + (−0.302 − 0.379i)9-s + (−0.00940 + 0.00452i)10-s + (−0.879 + 1.10i)11-s + (−0.156 + 0.686i)12-s + (0.172 − 0.216i)13-s + (−0.106 − 0.0823i)14-s + (0.0346 + 0.0434i)15-s + (−0.852 + 0.410i)16-s + (−0.0122 + 0.0534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.544 - 0.838i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.544 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748747 + 1.37937i\)
\(L(\frac12)\) \(\approx\) \(0.748747 + 1.37937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0412 - 2.64i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.118 - 0.149i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-1.11 - 0.539i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.156 - 0.0751i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.91 - 3.65i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (0.0503 - 0.220i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 + (-0.891 - 3.90i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.643 - 2.81i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 7.49T + 31T^{2} \)
37 \( 1 + (-0.461 + 2.02i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-0.210 - 0.101i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-9.99 + 4.81i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.33 - 7.94i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.63 - 11.5i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-3.22 + 1.55i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.269 + 1.18i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 + (0.892 + 3.90i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.34 - 2.93i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 8.60T + 79T^{2} \)
83 \( 1 + (-8.78 - 11.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.67 - 4.60i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 7.72T + 97T^{2} \)
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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

−10.88009448367815429595860406848, −9.668784741943648343410552675438, −9.171281992858759636946991558224, −8.168128261912515343696878983914, −7.68463426169737030522059012036, −6.49368440878864340649210466157, −5.40984959113739788251198123943, −4.17859894207618278652843212069, −3.02110036164686357438535111752, −2.34952434731580160788466869568, 0.798732248350476963924399306668, 2.25428435568055119776495036122, 3.34778451847590387824053068317, 4.83020972577230451397999264376, 5.74792943800538080058830385859, 6.74711870329820288907632724453, 7.77342144361405261251337481181, 8.442337802369645536657735945865, 9.520158644916403530826737933780, 10.32786534650346132713721131268

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