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Properties

Label 2-50-5.4-c5-0-0
Degree $2$
Conductor $50$
Sign $-0.894 - 0.447i$
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  − 4i·2-s + 26i·3-s − 16·4-s + 104·6-s − 22i·7-s + 64i·8-s − 433·9-s − 768·11-s − 416i·12-s + 46i·13-s − 88·14-s + 256·16-s + 378i·17-s + 1.73e3i·18-s − 1.10e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.66i·3-s − 0.5·4-s + 1.17·6-s − 0.169i·7-s + 0.353i·8-s − 1.78·9-s − 1.91·11-s − 0.833i·12-s + 0.0754i·13-s − 0.119·14-s + 0.250·16-s + 0.317i·17-s + 1.25i·18-s − 0.699·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.146232 + 0.619448i\)
\(L(\frac12)\) \(\approx\) \(0.146232 + 0.619448i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 \)
good3 \( 1 - 26iT - 243T^{2} \)
7 \( 1 + 22iT - 1.68e4T^{2} \)
11 \( 1 + 768T + 1.61e5T^{2} \)
13 \( 1 - 46iT - 3.71e5T^{2} \)
17 \( 1 - 378iT - 1.41e6T^{2} \)
19 \( 1 + 1.10e3T + 2.47e6T^{2} \)
23 \( 1 - 1.98e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.61e3T + 2.05e7T^{2} \)
31 \( 1 + 3.98e3T + 2.86e7T^{2} \)
37 \( 1 + 142iT - 6.93e7T^{2} \)
41 \( 1 - 1.54e3T + 1.15e8T^{2} \)
43 \( 1 - 5.02e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.47e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.41e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.83e4T + 7.14e8T^{2} \)
61 \( 1 - 5.52e3T + 8.44e8T^{2} \)
67 \( 1 + 2.47e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.23e4T + 1.80e9T^{2} \)
73 \( 1 - 5.21e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.96e4T + 3.07e9T^{2} \)
83 \( 1 - 5.98e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.76e4T + 5.58e9T^{2} \)
97 \( 1 + 1.44e5iT - 8.58e9T^{2} \)
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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

−15.23501730075735007855910143481, −13.93606359404150330918530790071, −12.64880755379914409746304691856, −10.99876659575571777670651588653, −10.46572395375783880175663740848, −9.445664976751786812730331439591, −8.118908283399359007773267981675, −5.45831892377739789670431493775, −4.32854548765431288603696569022, −2.86332319494168271911263162703, 0.30827538427490138079739408299, 2.44318283480698854920959325336, 5.30505788922073790936057385707, 6.60999869451165999819427217355, 7.69996137015797408150125367913, 8.539798430126657057518600147150, 10.54250197546764072898436812530, 12.22008494901334422419986718194, 13.05265918477547073785973615812, 13.82250123480464341421603471862

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