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Properties

Label 2-51-17.4-c5-0-13
Degree $2$
Conductor $51$
Sign $-0.978 + 0.206i$
Analytic cond. $8.17957$
Root an. cond. $2.85999$
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  − 8.22i·2-s + (6.36 − 6.36i)3-s − 35.5·4-s + (66.1 − 66.1i)5-s + (−52.3 − 52.3i)6-s + (13.8 + 13.8i)7-s + 29.3i·8-s − 81i·9-s + (−543. − 543. i)10-s + (169. + 169. i)11-s + (−226. + 226. i)12-s − 213.·13-s + (113. − 113. i)14-s − 842. i·15-s − 896.·16-s + (386. + 1.12e3i)17-s + ⋯
L(s)  = 1  − 1.45i·2-s + (0.408 − 0.408i)3-s − 1.11·4-s + (1.18 − 1.18i)5-s + (−0.593 − 0.593i)6-s + (0.106 + 0.106i)7-s + 0.162i·8-s − 0.333i·9-s + (−1.72 − 1.72i)10-s + (0.421 + 0.421i)11-s + (−0.453 + 0.453i)12-s − 0.351·13-s + (0.155 − 0.155i)14-s − 0.966i·15-s − 0.875·16-s + (0.324 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.978 + 0.206i$
Analytic conductor: \(8.17957\)
Root analytic conductor: \(2.85999\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :5/2),\ -0.978 + 0.206i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.227722 - 2.18688i\)
\(L(\frac12)\) \(\approx\) \(0.227722 - 2.18688i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.36 + 6.36i)T \)
17 \( 1 + (-386. - 1.12e3i)T \)
good2 \( 1 + 8.22iT - 32T^{2} \)
5 \( 1 + (-66.1 + 66.1i)T - 3.12e3iT^{2} \)
7 \( 1 + (-13.8 - 13.8i)T + 1.68e4iT^{2} \)
11 \( 1 + (-169. - 169. i)T + 1.61e5iT^{2} \)
13 \( 1 + 213.T + 3.71e5T^{2} \)
19 \( 1 - 2.70e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.22e3 + 1.22e3i)T + 6.43e6iT^{2} \)
29 \( 1 + (-371. + 371. i)T - 2.05e7iT^{2} \)
31 \( 1 + (-6.14e3 + 6.14e3i)T - 2.86e7iT^{2} \)
37 \( 1 + (-5.41e3 + 5.41e3i)T - 6.93e7iT^{2} \)
41 \( 1 + (3.65e3 + 3.65e3i)T + 1.15e8iT^{2} \)
43 \( 1 - 1.86e4iT - 1.47e8T^{2} \)
47 \( 1 - 8.80e3T + 2.29e8T^{2} \)
53 \( 1 - 3.08e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.47e4iT - 7.14e8T^{2} \)
61 \( 1 + (-3.00e3 - 3.00e3i)T + 8.44e8iT^{2} \)
67 \( 1 - 6.30e3T + 1.35e9T^{2} \)
71 \( 1 + (-3.19e4 + 3.19e4i)T - 1.80e9iT^{2} \)
73 \( 1 + (4.60e4 - 4.60e4i)T - 2.07e9iT^{2} \)
79 \( 1 + (5.29e4 + 5.29e4i)T + 3.07e9iT^{2} \)
83 \( 1 - 6.65e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.02e5T + 5.58e9T^{2} \)
97 \( 1 + (5.56e4 - 5.56e4i)T - 8.58e9iT^{2} \)
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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

−13.49152639177070163803536814166, −12.62018745554998520212412561440, −12.00864820879970112493671059809, −10.19043206932325291654958090326, −9.518071061843739311042739597477, −8.272589274000157592052182395076, −6.00862840300960251065154508626, −4.20227034344810206594829004380, −2.18661646282705965943445637237, −1.22224758877212241621130032194, 2.72961838650325859557103659729, 5.06253600553020176389139973191, 6.39870241577858520930876712394, 7.30214324989191385797369394225, 8.892069938679966758433565746741, 9.956595628924929786789046742682, 11.30228469197657555380118047739, 13.72710220859105052669957783095, 14.00024441689773600462010120283, 15.03414203526464719142309983682

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