↑

Properties

Label 2-50-5.2-c4-0-2
Degree $2$
Conductor $50$
Sign $0.130 - 0.991i$
Analytic cond. $5.16849$
Root an. cond. $2.27343$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)2-s + (−0.123 + 0.123i)3-s + 8i·4-s − 0.494·6-s + (60.4 + 60.4i)7-s + (−16 + 16i)8-s + 80.9i·9-s − 136.·11-s + (−0.989 − 0.989i)12-s + (146. − 146. i)13-s + 241. i·14-s − 64·16-s + (100. + 100. i)17-s + (−161. + 161. i)18-s − 275. i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.0137 + 0.0137i)3-s + 0.5i·4-s − 0.0137·6-s + (1.23 + 1.23i)7-s + (−0.250 + 0.250i)8-s + 0.999i·9-s − 1.12·11-s + (−0.00687 − 0.00687i)12-s + (0.865 − 0.865i)13-s + 1.23i·14-s − 0.250·16-s + (0.349 + 0.349i)17-s + (−0.499 + 0.499i)18-s − 0.763i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(5.16849\)
Root analytic conductor: \(2.27343\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :2),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.53800 + 1.34846i\)
\(L(\frac12)\) \(\approx\) \(1.53800 + 1.34846i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
5 \( 1 \)
good3 \( 1 + (0.123 - 0.123i)T - 81iT^{2} \)
7 \( 1 + (-60.4 - 60.4i)T + 2.40e3iT^{2} \)
11 \( 1 + 136.T + 1.46e4T^{2} \)
13 \( 1 + (-146. + 146. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-100. - 100. i)T + 8.35e4iT^{2} \)
19 \( 1 + 275. iT - 1.30e5T^{2} \)
23 \( 1 + (-126. + 126. i)T - 2.79e5iT^{2} \)
29 \( 1 + 1.43e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.28e3T + 9.23e5T^{2} \)
37 \( 1 + (1.09e3 + 1.09e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.09e3T + 2.82e6T^{2} \)
43 \( 1 + (150. - 150. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.62e3 - 1.62e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-228. + 228. i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.42e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.01e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.92e3 - 1.92e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.99e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.57e3 + 3.57e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 1.91e3iT - 3.89e7T^{2} \)
83 \( 1 + (918. - 918. i)T - 4.74e7iT^{2} \)
89 \( 1 + 9.26e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.87e3 + 2.87e3i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.33728344994140633611915572738, −13.92866619214393369097456014433, −12.97720346771342786479446890603, −11.66245243443022661520200983933, −10.57068529367832599554776916898, −8.479465908712631925187128226035, −7.85942691616140678571735829395, −5.75963078693626275491334000798, −4.87872062741949682489842243935, −2.48394943210630870959159368334, 1.27565431687900062895696871103, 3.65995061175126131598930354229, 5.05873866618685460622333706827, 6.88305156405962330942798731650, 8.396180034018160416355433317141, 10.12440326558492810452050979879, 11.08510541327070036911155845999, 12.13789090012936423040330543297, 13.56734872707302372671792026428, 14.24840808756204280206760582047

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