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Properties

Label 2-637-49.43-c1-0-30
Degree $2$
Conductor $637$
Sign $-0.446 - 0.894i$
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  + (−1.57 + 1.96i)2-s + (2.34 + 1.13i)3-s + (−0.967 − 4.23i)4-s + (3.79 + 1.82i)5-s + (−5.91 + 2.84i)6-s + (0.107 − 2.64i)7-s + (5.32 + 2.56i)8-s + (2.35 + 2.95i)9-s + (−9.55 + 4.60i)10-s + (1.87 − 2.34i)11-s + (2.51 − 11.0i)12-s + (0.623 − 0.781i)13-s + (5.03 + 4.36i)14-s + (6.83 + 8.57i)15-s + (−5.59 + 2.69i)16-s + (−0.290 + 1.27i)17-s + ⋯
L(s)  = 1  + (−1.11 + 1.39i)2-s + (1.35 + 0.652i)3-s + (−0.483 − 2.11i)4-s + (1.69 + 0.816i)5-s + (−2.41 + 1.16i)6-s + (0.0407 − 0.999i)7-s + (1.88 + 0.907i)8-s + (0.786 + 0.985i)9-s + (−3.02 + 1.45i)10-s + (0.565 − 0.708i)11-s + (0.727 − 3.18i)12-s + (0.172 − 0.216i)13-s + (1.34 + 1.16i)14-s + (1.76 + 2.21i)15-s + (−1.39 + 0.673i)16-s + (−0.0705 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.446 - 0.894i$
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: \((2,\ 637,\ (\ :1/2),\ -0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907519 + 1.46791i\)
\(L(\frac12)\) \(\approx\) \(0.907519 + 1.46791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.107 + 2.64i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (1.57 - 1.96i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-2.34 - 1.13i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-3.79 - 1.82i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.87 + 2.34i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (0.290 - 1.27i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.95T + 19T^{2} \)
23 \( 1 + (-0.0448 - 0.196i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-0.542 + 2.37i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.570T + 31T^{2} \)
37 \( 1 + (2.37 - 10.4i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.77 - 2.29i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-8.15 + 3.92i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (3.64 - 4.56i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.570 - 2.49i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (10.1 - 4.90i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-1.72 + 7.56i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 5.63T + 67T^{2} \)
71 \( 1 + (2.86 + 12.5i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.25 + 6.59i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (2.67 + 3.34i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.566 - 0.710i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 3.36T + 97T^{2} \)
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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

−10.43030699452186515560066507691, −9.688661904223819045233177831592, −9.071991995028455913958705336272, −8.394954629627214780677430094285, −7.47716228859430888303884270445, −6.43143286352969548111127100754, −6.02864299916426041640721153616, −4.48139854667831817201652748001, −3.05785128940968974121536955884, −1.62430815125603729516706195216, 1.49515712852206187227883479240, 2.10916595783391859932835975643, 2.69305758394391637317078570639, 4.27228797258484144935308169443, 5.84552235733958140415043254363, 7.10369145719928097285525237757, 8.412475662139664525793005576625, 8.853356502228210151266811175288, 9.275298491580688555956748343577, 9.913997946184568984542272856421

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