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Properties

Label 2-637-49.8-c1-0-36
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: $\chi_{637} (547, \cdot )$
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

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

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