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Properties

Label 2-7e2-7.2-c5-0-13
Degree $2$
Conductor $49$
Sign $-0.947 - 0.318i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.90 − 6.76i)2-s + (−11.7 − 20.3i)3-s + (−14.5 − 25.1i)4-s + (37.1 − 64.3i)5-s − 183.·6-s + 22.8·8-s + (−155. + 268. i)9-s + (−290. − 502. i)10-s + (212. + 367. i)11-s + (−341. + 592. i)12-s − 252.·13-s − 1.74e3·15-s + (554. − 960. i)16-s + (552. + 956. i)17-s + (1.21e3 + 2.09e3i)18-s + (3.23 − 5.60i)19-s + ⋯
L(s)  = 1  + (0.690 − 1.19i)2-s + (−0.754 − 1.30i)3-s + (−0.454 − 0.786i)4-s + (0.664 − 1.15i)5-s − 2.08·6-s + 0.126·8-s + (−0.638 + 1.10i)9-s + (−0.917 − 1.58i)10-s + (0.528 + 0.915i)11-s + (−0.685 + 1.18i)12-s − 0.413·13-s − 2.00·15-s + (0.541 − 0.937i)16-s + (0.463 + 0.802i)17-s + (0.881 + 1.52i)18-s + (0.00205 − 0.00356i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.947 - 0.318i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.331300 + 2.02573i\)
\(L(\frac12)\) \(\approx\) \(0.331300 + 2.02573i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-3.90 + 6.76i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (11.7 + 20.3i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-37.1 + 64.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-212. - 367. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 252.T + 3.71e5T^{2} \)
17 \( 1 + (-552. - 956. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-3.23 + 5.60i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.80e3 + 3.12e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 5.00e3T + 2.05e7T^{2} \)
31 \( 1 + (1.41e3 + 2.44e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.02e3 + 1.77e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 9.39e3T + 1.15e8T^{2} \)
43 \( 1 - 1.03e4T + 1.47e8T^{2} \)
47 \( 1 + (8.51e3 - 1.47e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.97e4 - 3.42e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.69e4 + 2.94e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.41e4 - 2.45e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.80e4 + 4.85e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 1.55e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.91e4 - 6.77e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.26e4 + 3.92e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.38e3T + 3.93e9T^{2} \)
89 \( 1 + (-3.44e4 + 5.96e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.08e5T + 8.58e9T^{2} \)
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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

−13.22827555724588715841668644051, −12.60869044177866024989671828475, −12.17877900957590538010028864274, −10.85990767642170353774269852147, −9.381993051123879888130872564287, −7.49889964007213940707466960142, −5.87360222460017352265734089575, −4.55618893221208174406971048281, −2.04075490907682545513325787181, −1.03031907024905714493777001066, 3.55882501981732478680781781255, 5.18780835087480420964670398745, 6.01900307272834309488203636124, 7.24420777200368462948541240464, 9.399210986151361845546463598291, 10.54614940909286375196430180052, 11.45949511801910578425509376074, 13.52038704888575235264212918300, 14.48329159651383931637094889700, 15.15847840265542252895025179137

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