Properties

Label 2-637-49.4-c1-0-30
Degree $2$
Conductor $637$
Sign $-0.155 - 0.987i$
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  + (2.14 + 1.46i)2-s + (0.378 + 0.350i)3-s + (1.74 + 4.43i)4-s + (0.273 − 0.0844i)5-s + (0.298 + 1.30i)6-s + (2.61 − 0.388i)7-s + (−1.59 + 7.00i)8-s + (−0.204 − 2.72i)9-s + (0.712 + 0.219i)10-s + (−0.103 + 1.38i)11-s + (−0.897 + 2.28i)12-s + (0.900 − 0.433i)13-s + (6.19 + 2.99i)14-s + (0.133 + 0.0641i)15-s + (−6.71 + 6.23i)16-s + (−1.96 + 0.296i)17-s + ⋯
L(s)  = 1  + (1.51 + 1.03i)2-s + (0.218 + 0.202i)3-s + (0.870 + 2.21i)4-s + (0.122 − 0.0377i)5-s + (0.121 + 0.534i)6-s + (0.989 − 0.146i)7-s + (−0.565 + 2.47i)8-s + (−0.0680 − 0.908i)9-s + (0.225 + 0.0694i)10-s + (−0.0311 + 0.416i)11-s + (−0.259 + 0.660i)12-s + (0.249 − 0.120i)13-s + (1.65 + 0.801i)14-s + (0.0344 + 0.0165i)15-s + (−1.67 + 1.55i)16-s + (−0.476 + 0.0718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50407 + 2.92845i\)
\(L(\frac12)\) \(\approx\) \(2.50407 + 2.92845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.61 + 0.388i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (-2.14 - 1.46i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (-0.378 - 0.350i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-0.273 + 0.0844i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.103 - 1.38i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (1.96 - 0.296i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (2.10 + 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.82 + 1.02i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (0.519 + 0.651i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-1.15 + 1.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.10 - 7.90i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-2.75 + 12.0i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-1.89 - 8.32i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (6.91 + 4.71i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (0.798 + 2.03i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-12.8 - 3.96i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (-0.510 + 1.30i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-0.370 + 0.641i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.60 + 7.02i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.26 - 4.27i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-7.52 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.552 + 0.266i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.621 - 8.29i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + 7.36T + 97T^{2} \)
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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

−11.23728843591815259104000494711, −9.913671311002828048926722343366, −8.674014704839369812562051079332, −7.990668754683307453239888294901, −7.00527638805229068555735255255, −6.23679524353910000429918244491, −5.29606365266326535470993060156, −4.36045007276717975763033325720, −3.71287864376202313834355929610, −2.27801354381710787247485834873, 1.71209163987499488482036820174, 2.36619086730651323884229228314, 3.75960724994039050600962953944, 4.59537221841711488831581043351, 5.52767718159501697817166745213, 6.25600677905114149679596433711, 7.70478424073229762563462649758, 8.555970732940790321669466773407, 9.974140342227117944918002234875, 10.68835353270771376142622818674

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