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Properties

Label 2-637-49.43-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.712 - 0.701i$
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  + (−0.807 + 1.01i)2-s + (−2.74 − 1.32i)3-s + (0.0716 + 0.313i)4-s + (2.02 + 0.975i)5-s + (3.55 − 1.71i)6-s + (0.327 + 2.62i)7-s + (−2.70 − 1.30i)8-s + (3.90 + 4.90i)9-s + (−2.62 + 1.26i)10-s + (0.920 − 1.15i)11-s + (0.218 − 0.955i)12-s + (0.623 − 0.781i)13-s + (−2.92 − 1.78i)14-s + (−4.26 − 5.34i)15-s + (2.93 − 1.41i)16-s + (−0.687 + 3.01i)17-s + ⋯
L(s)  = 1  + (−0.571 + 0.716i)2-s + (−1.58 − 0.762i)3-s + (0.0358 + 0.156i)4-s + (0.905 + 0.436i)5-s + (1.45 − 0.698i)6-s + (0.123 + 0.992i)7-s + (−0.958 − 0.461i)8-s + (1.30 + 1.63i)9-s + (−0.829 + 0.399i)10-s + (0.277 − 0.347i)11-s + (0.0629 − 0.275i)12-s + (0.172 − 0.216i)13-s + (−0.781 − 0.478i)14-s + (−1.10 − 1.38i)15-s + (0.732 − 0.352i)16-s + (−0.166 + 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.712 - 0.701i$
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: \((2,\ 637,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242739 + 0.592405i\)
\(L(\frac12)\) \(\approx\) \(0.242739 + 0.592405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.327 - 2.62i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.807 - 1.01i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (2.74 + 1.32i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-2.02 - 0.975i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-0.920 + 1.15i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (0.687 - 3.01i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 8.01T + 19T^{2} \)
23 \( 1 + (0.871 + 3.81i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (2.13 - 9.34i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.571T + 31T^{2} \)
37 \( 1 + (1.03 - 4.52i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.62 - 2.22i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (6.68 - 3.21i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (4.13 - 5.17i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.0492 - 0.215i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (9.48 - 4.56i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.08 - 9.13i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 2.01T + 67T^{2} \)
71 \( 1 + (3.29 + 14.4i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-10.4 - 13.0i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 4.15T + 79T^{2} \)
83 \( 1 + (1.02 + 1.28i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.42 + 1.78i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 2.98T + 97T^{2} \)
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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

−11.00667542836220340018763045582, −10.02779540947352751937476494818, −9.098096668756150439498375883327, −8.087043810807883564891671326020, −7.15494185609952618567198329131, −6.30249111463822672988912607450, −5.94235266896570125781191534425, −5.08428437929762464720506213553, −3.01164730013598374709252133489, −1.43687249102570410770020067002, 0.56245995614193126502668173626, 1.66679220538331700345419955458, 3.67403672724680776414917362792, 4.93525474035229886946324226507, 5.54354001447526137587584534268, 6.38946084655036641566254085753, 7.49268453756241590439348854507, 9.253811774955405113943147472991, 9.759287530890713944014567153779, 10.08536210105300747536349784661

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