Properties

Label 2-7e2-49.25-c5-0-2
Degree $2$
Conductor $49$
Sign $-0.0549 - 0.998i$
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  + (−0.768 + 0.237i)2-s + (−8.72 − 22.2i)3-s + (−25.9 + 17.6i)4-s + (28.5 + 4.30i)5-s + (11.9 + 15.0i)6-s + (−107. − 71.7i)7-s + (31.7 − 39.8i)8-s + (−240. + 222. i)9-s + (−22.9 + 3.46i)10-s + (432. + 400. i)11-s + (618. + 421. i)12-s + (−206. + 904. i)13-s + (100. + 29.6i)14-s + (−153. − 672. i)15-s + (351. − 895. i)16-s + (−156. + 2.09e3i)17-s + ⋯
L(s)  = 1  + (−0.135 + 0.0419i)2-s + (−0.559 − 1.42i)3-s + (−0.809 + 0.551i)4-s + (0.510 + 0.0770i)5-s + (0.135 + 0.170i)6-s + (−0.832 − 0.553i)7-s + (0.175 − 0.220i)8-s + (−0.988 + 0.917i)9-s + (−0.0726 + 0.0109i)10-s + (1.07 + 0.998i)11-s + (1.24 + 0.845i)12-s + (−0.338 + 1.48i)13-s + (0.136 + 0.0403i)14-s + (−0.176 − 0.771i)15-s + (0.343 − 0.874i)16-s + (−0.131 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0549 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0549 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.0549 - 0.998i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.0549 - 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.282010 + 0.297970i\)
\(L(\frac12)\) \(\approx\) \(0.282010 + 0.297970i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (107. + 71.7i)T \)
good2 \( 1 + (0.768 - 0.237i)T + (26.4 - 18.0i)T^{2} \)
3 \( 1 + (8.72 + 22.2i)T + (-178. + 165. i)T^{2} \)
5 \( 1 + (-28.5 - 4.30i)T + (2.98e3 + 921. i)T^{2} \)
11 \( 1 + (-432. - 400. i)T + (1.20e4 + 1.60e5i)T^{2} \)
13 \( 1 + (206. - 904. i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + (156. - 2.09e3i)T + (-1.40e6 - 2.11e5i)T^{2} \)
19 \( 1 + (953. + 1.65e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (24.4 + 326. i)T + (-6.36e6 + 9.59e5i)T^{2} \)
29 \( 1 + (-3.83e3 + 1.84e3i)T + (1.27e7 - 1.60e7i)T^{2} \)
31 \( 1 + (3.56e3 - 6.16e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.54e3 + 3.77e3i)T + (2.53e7 + 6.45e7i)T^{2} \)
41 \( 1 + (8.49e3 - 1.06e4i)T + (-2.57e7 - 1.12e8i)T^{2} \)
43 \( 1 + (2.37e3 + 2.97e3i)T + (-3.27e7 + 1.43e8i)T^{2} \)
47 \( 1 + (1.46e4 - 4.53e3i)T + (1.89e8 - 1.29e8i)T^{2} \)
53 \( 1 + (-3.38e3 + 2.30e3i)T + (1.52e8 - 3.89e8i)T^{2} \)
59 \( 1 + (-3.52e4 + 5.31e3i)T + (6.83e8 - 2.10e8i)T^{2} \)
61 \( 1 + (-1.24e4 - 8.46e3i)T + (3.08e8 + 7.86e8i)T^{2} \)
67 \( 1 + (2.45e4 - 4.24e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.52e4 + 1.21e4i)T + (1.12e9 + 1.41e9i)T^{2} \)
73 \( 1 + (5.66e4 + 1.74e4i)T + (1.71e9 + 1.16e9i)T^{2} \)
79 \( 1 + (-779. - 1.34e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-824. - 3.61e3i)T + (-3.54e9 + 1.70e9i)T^{2} \)
89 \( 1 + (1.80e4 - 1.67e4i)T + (4.17e8 - 5.56e9i)T^{2} \)
97 \( 1 - 3.71e4T + 8.58e9T^{2} \)
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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

−14.45829454034823536846107797122, −13.42592190371598127809047883083, −12.71568750782748401720710174246, −11.83998589501374982734677370864, −9.986482948833307444146043740477, −8.744228891489194903309718356177, −7.05971101430290719808267328526, −6.46580606349733893252682319291, −4.23080505994607459204424958433, −1.67808477191005070143749659636, 0.24390201042765851654518118221, 3.55100159833810699119503314356, 5.20361301737259664672035939224, 5.97705877213865634283587550626, 8.777116352144559017284581113567, 9.686546443807507549240244902776, 10.31047371440068457604808899257, 11.71969051609266817913866801866, 13.29903890061593248906822402913, 14.47894674658156595875197290423

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