Properties

Label 2-7e2-49.25-c5-0-13
Degree $2$
Conductor $49$
Sign $-0.135 + 0.990i$
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  + (−3.14 + 0.970i)2-s + (−3.81 − 9.71i)3-s + (−17.4 + 11.9i)4-s + (67.5 + 10.1i)5-s + (21.4 + 26.8i)6-s + (−81.7 + 100. i)7-s + (109. − 136. i)8-s + (98.3 − 91.2i)9-s + (−222. + 33.5i)10-s + (−557. − 517. i)11-s + (182. + 124. i)12-s + (36.0 − 158. i)13-s + (159. − 395. i)14-s + (−158. − 695. i)15-s + (36.9 − 94.0i)16-s + (121. − 1.62e3i)17-s + ⋯
L(s)  = 1  + (−0.556 + 0.171i)2-s + (−0.244 − 0.622i)3-s + (−0.546 + 0.372i)4-s + (1.20 + 0.182i)5-s + (0.242 + 0.304i)6-s + (−0.630 + 0.776i)7-s + (0.602 − 0.755i)8-s + (0.404 − 0.375i)9-s + (−0.703 + 0.106i)10-s + (−1.39 − 1.28i)11-s + (0.365 + 0.249i)12-s + (0.0592 − 0.259i)13-s + (0.217 − 0.539i)14-s + (−0.182 − 0.797i)15-s + (0.0360 − 0.0918i)16-s + (0.102 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.135 + 0.990i$
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: \((2,\ 49,\ (\ :5/2),\ -0.135 + 0.990i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.514443 - 0.589381i\)
\(L(\frac12)\) \(\approx\) \(0.514443 - 0.589381i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (81.7 - 100. i)T \)
good2 \( 1 + (3.14 - 0.970i)T + (26.4 - 18.0i)T^{2} \)
3 \( 1 + (3.81 + 9.71i)T + (-178. + 165. i)T^{2} \)
5 \( 1 + (-67.5 - 10.1i)T + (2.98e3 + 921. i)T^{2} \)
11 \( 1 + (557. + 517. i)T + (1.20e4 + 1.60e5i)T^{2} \)
13 \( 1 + (-36.0 + 158. i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + (-121. + 1.62e3i)T + (-1.40e6 - 2.11e5i)T^{2} \)
19 \( 1 + (-257. - 446. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (249. + 3.32e3i)T + (-6.36e6 + 9.59e5i)T^{2} \)
29 \( 1 + (-3.84e3 + 1.85e3i)T + (1.27e7 - 1.60e7i)T^{2} \)
31 \( 1 + (2.37e3 - 4.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.05e3 - 3.44e3i)T + (2.53e7 + 6.45e7i)T^{2} \)
41 \( 1 + (838. - 1.05e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
43 \( 1 + (6.60e3 + 8.27e3i)T + (-3.27e7 + 1.43e8i)T^{2} \)
47 \( 1 + (-1.05e4 + 3.26e3i)T + (1.89e8 - 1.29e8i)T^{2} \)
53 \( 1 + (7.54e3 - 5.14e3i)T + (1.52e8 - 3.89e8i)T^{2} \)
59 \( 1 + (-3.76e4 + 5.67e3i)T + (6.83e8 - 2.10e8i)T^{2} \)
61 \( 1 + (3.39e4 + 2.31e4i)T + (3.08e8 + 7.86e8i)T^{2} \)
67 \( 1 + (8.15e3 - 1.41e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (6.54e4 + 3.14e4i)T + (1.12e9 + 1.41e9i)T^{2} \)
73 \( 1 + (-2.75e4 - 8.48e3i)T + (1.71e9 + 1.16e9i)T^{2} \)
79 \( 1 + (-3.43e4 - 5.95e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.69e4 - 1.18e5i)T + (-3.54e9 + 1.70e9i)T^{2} \)
89 \( 1 + (1.26e4 - 1.17e4i)T + (4.17e8 - 5.56e9i)T^{2} \)
97 \( 1 + 3.92e4T + 8.58e9T^{2} \)
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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

−13.85640395126304234719890184684, −13.21314530056168944541440464085, −12.25786636946508707411705794036, −10.35682541461703502849817690083, −9.406225108537649444179872994510, −8.204833937215682903007937327953, −6.66040248539698948732076528069, −5.44577083103592613124953780477, −2.81023160924011501068541998005, −0.50763853660104686257115576039, 1.75069479359874275268767536857, 4.48003395970039809881499922602, 5.66429766893269423619772840943, 7.60916704266531689506943921860, 9.433718471262146810911850764683, 10.09127392543231758581897740209, 10.63761672248036684763150263882, 13.03101427185707062267820450132, 13.48150024868228315647061300031, 14.96943182844140875200450828088

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