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Properties

Label 2-637-7.4-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.701 + 0.712i$
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  + (0.906 + 1.57i)2-s + (−1.55 + 2.68i)3-s + (−0.644 + 1.11i)4-s + (1.40 + 2.43i)5-s − 5.62·6-s + 1.28·8-s + (−3.31 − 5.73i)9-s + (−2.55 + 4.41i)10-s + (−1.55 + 2.68i)11-s + (−2 − 3.46i)12-s − 13-s − 8.72·15-s + (2.45 + 4.25i)16-s + (−0.262 + 0.454i)17-s + (6.00 − 10.4i)18-s + (0.406 + 0.704i)19-s + ⋯
L(s)  = 1  + (0.641 + 1.11i)2-s + (−0.895 + 1.55i)3-s + (−0.322 + 0.558i)4-s + (0.629 + 1.08i)5-s − 2.29·6-s + 0.455·8-s + (−1.10 − 1.91i)9-s + (−0.806 + 1.39i)10-s + (−0.467 + 0.810i)11-s + (−0.577 − 0.999i)12-s − 0.277·13-s − 2.25·15-s + (0.614 + 1.06i)16-s + (−0.0635 + 0.110i)17-s + (1.41 − 2.45i)18-s + (0.0933 + 0.161i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629016 - 1.50093i\)
\(L(\frac12)\) \(\approx\) \(0.629016 - 1.50093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + (-0.906 - 1.57i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.55 - 2.68i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.40 - 2.43i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.55 - 2.68i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.262 - 0.454i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.406 - 0.704i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.66 + 6.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + (-0.695 + 1.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.07 - 5.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.20T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 + (2.98 + 5.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.24 + 2.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.23 - 3.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.01 - 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.72T + 71T^{2} \)
73 \( 1 + (1.17 - 2.02i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.77 - 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + (5.32 + 9.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.18T + 97T^{2} \)
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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

−10.77075359297496732623821858120, −10.22229773079471605947478478593, −9.860432983980560533898650850793, −8.430742380790530112496929463640, −7.14468163194066106584716347651, −6.32682940953795952650042079449, −5.78675076298302276214829733258, −4.75908726347504027279103993852, −4.21252836670046350555412295825, −2.70773884382702482345612865720, 0.833195135585768240361321609726, 1.75351923697494255766619066191, 2.84483254243896179756225455504, 4.57077906792325200306757563257, 5.46355984273638963438637246969, 6.07793994704081283157373153652, 7.40321033069247233674675982281, 8.100384631701187024232234947553, 9.305990736981342957082896582443, 10.47492187228823632483050621938

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