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Properties

Label 2-637-7.4-c1-0-34
Degree $2$
Conductor $637$
Sign $0.605 + 0.795i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.605 + 0.795i$
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: \((2,\ 637,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24672 - 1.11376i\)
\(L(\frac12)\) \(\approx\) \(2.24672 - 1.11376i\)
\(L(\frac{3}{2})\) 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

−10.37876055836203923103595388226, −8.992606319931099622570900445361, −8.162089104575777730861866319117, −7.88758777095419973135066474739, −6.89161818564958277400499482374, −6.27053896757488271903106297967, −4.99800881860522766095538992223, −4.12297974696752882722800629974, −2.49673093082425735707832027179, −1.09346963801855202203564243847, 2.39862102036528839532857839965, 3.28986930422043142684618274531, 3.69222057699002373465858740686, 4.62099964575788660596082373368, 5.78762112016379350713758132885, 7.47951890334071729050266971533, 8.209673447296765443680153720094, 9.278278321395144999336691334686, 10.31147211844883902401489804277, 10.61332486398142710473460845019

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