Properties

Label 2-50-25.6-c5-0-3
Degree $2$
Conductor $50$
Sign $-0.882 - 0.471i$
Analytic cond. $8.01919$
Root an. cond. $2.83181$
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  + (1.23 + 3.80i)2-s + (8.60 − 6.25i)3-s + (−12.9 + 9.40i)4-s + (−38.0 + 40.9i)5-s + (34.4 + 25.0i)6-s − 106.·7-s + (−51.7 − 37.6i)8-s + (−40.1 + 123. i)9-s + (−202. − 93.9i)10-s + (155. + 479. i)11-s + (−52.5 + 161. i)12-s + (−61.5 + 189. i)13-s + (−131. − 405. i)14-s + (−70.9 + 590. i)15-s + (79.1 − 243. i)16-s + (−415. − 302. i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.552 − 0.401i)3-s + (−0.404 + 0.293i)4-s + (−0.680 + 0.733i)5-s + (0.390 + 0.283i)6-s − 0.822·7-s + (−0.286 − 0.207i)8-s + (−0.165 + 0.508i)9-s + (−0.641 − 0.297i)10-s + (0.388 + 1.19i)11-s + (−0.105 + 0.324i)12-s + (−0.101 + 0.311i)13-s + (−0.179 − 0.553i)14-s + (−0.0814 + 0.677i)15-s + (0.0772 − 0.237i)16-s + (−0.348 − 0.253i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.882 - 0.471i$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ -0.882 - 0.471i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.294074 + 1.17484i\)
\(L(\frac12)\) \(\approx\) \(0.294074 + 1.17484i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 3.80i)T \)
5 \( 1 + (38.0 - 40.9i)T \)
good3 \( 1 + (-8.60 + 6.25i)T + (75.0 - 231. i)T^{2} \)
7 \( 1 + 106.T + 1.68e4T^{2} \)
11 \( 1 + (-155. - 479. i)T + (-1.30e5 + 9.46e4i)T^{2} \)
13 \( 1 + (61.5 - 189. i)T + (-3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (415. + 302. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (1.27e3 + 923. i)T + (7.65e5 + 2.35e6i)T^{2} \)
23 \( 1 + (-418. - 1.28e3i)T + (-5.20e6 + 3.78e6i)T^{2} \)
29 \( 1 + (-3.91e3 + 2.84e3i)T + (6.33e6 - 1.95e7i)T^{2} \)
31 \( 1 + (-5.82e3 - 4.23e3i)T + (8.84e6 + 2.72e7i)T^{2} \)
37 \( 1 + (-43.3 + 133. i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (3.46e3 - 1.06e4i)T + (-9.37e7 - 6.80e7i)T^{2} \)
43 \( 1 - 6.84e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.00e4 + 7.26e3i)T + (7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (-2.24e4 + 1.62e4i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (1.33e4 - 4.09e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-4.18e3 - 1.28e4i)T + (-6.83e8 + 4.96e8i)T^{2} \)
67 \( 1 + (4.95e4 + 3.59e4i)T + (4.17e8 + 1.28e9i)T^{2} \)
71 \( 1 + (1.16e4 - 8.47e3i)T + (5.57e8 - 1.71e9i)T^{2} \)
73 \( 1 + (-2.49e4 - 7.69e4i)T + (-1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (6.75e4 - 4.90e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (2.80e4 + 2.03e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + (1.70e4 + 5.26e4i)T + (-4.51e9 + 3.28e9i)T^{2} \)
97 \( 1 + (-3.47e4 + 2.52e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−15.04087845810234775725871702264, −14.01918859855524191035282423573, −12.99727656608798418777730628669, −11.77901051056052042177473186849, −10.14699847196536521148928757202, −8.684116618596388050511968825861, −7.36707280390930826700538220109, −6.59925365513201426212438563172, −4.42590193827876770518077243431, −2.71347297683121664282172392261, 0.53775318021160487944068892785, 3.13784345041066502496752835595, 4.21208733255306631652829406207, 6.14968795555482694376857894881, 8.391478023396951916234530692234, 9.147050102221988282630277211625, 10.50581498773276479619660755042, 11.88871244804013143790272176930, 12.77960479225636864749235932219, 13.95752725075076393714391967149

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