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Properties

Label 2-116-29.28-c5-0-6
Degree $2$
Conductor $116$
Sign $0.919 + 0.393i$
Analytic cond. $18.6045$
Root an. cond. $4.31329$
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  − 13.6i·3-s + 81.0·5-s + 136.·7-s + 55.6·9-s + 588. i·11-s − 150.·13-s − 1.10e3i·15-s + 1.08e3i·17-s + 1.18e3i·19-s − 1.87e3i·21-s + 3.80e3·23-s + 3.43e3·25-s − 4.08e3i·27-s + (−4.16e3 − 1.78e3i)29-s − 9.24e3i·31-s + ⋯
L(s)  = 1  − 0.878i·3-s + 1.44·5-s + 1.05·7-s + 0.228·9-s + 1.46i·11-s − 0.247·13-s − 1.27i·15-s + 0.909i·17-s + 0.755i·19-s − 0.925i·21-s + 1.49·23-s + 1.10·25-s − 1.07i·27-s + (−0.919 − 0.393i)29-s − 1.72i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(18.6045\)
Root analytic conductor: \(4.31329\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :5/2),\ 0.919 + 0.393i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.812363402\)
\(L(\frac12)\) \(\approx\) \(2.812363402\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (4.16e3 + 1.78e3i)T \)
good3 \( 1 + 13.6iT - 243T^{2} \)
5 \( 1 - 81.0T + 3.12e3T^{2} \)
7 \( 1 - 136.T + 1.68e4T^{2} \)
11 \( 1 - 588. iT - 1.61e5T^{2} \)
13 \( 1 + 150.T + 3.71e5T^{2} \)
17 \( 1 - 1.08e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.18e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.80e3T + 6.43e6T^{2} \)
31 \( 1 + 9.24e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.38e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.84e4iT - 1.15e8T^{2} \)
43 \( 1 + 8.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.24e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.52e4T + 4.18e8T^{2} \)
59 \( 1 - 2.42e3T + 7.14e8T^{2} \)
61 \( 1 + 1.23e4iT - 8.44e8T^{2} \)
67 \( 1 - 7.40e3T + 1.35e9T^{2} \)
71 \( 1 + 5.59e4T + 1.80e9T^{2} \)
73 \( 1 - 2.50e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.91e4iT - 3.07e9T^{2} \)
83 \( 1 + 9.57e4T + 3.93e9T^{2} \)
89 \( 1 + 1.08e5iT - 5.58e9T^{2} \)
97 \( 1 - 4.25e4iT - 8.58e9T^{2} \)
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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

−12.87783392162784271522721970642, −11.65676177591917288488458034234, −10.28756494560304724555513767231, −9.493267116665466830710136219012, −8.003789574749981153986396023648, −7.02285967058597619259961602360, −5.84647849888422278164830849805, −4.58950085031325273629933550605, −2.08096565505199004525034624291, −1.52238717507895801445128800171, 1.30631709672521660400141548333, 2.98964305856872456049951758589, 4.82627844063987212208063754902, 5.53739042066801202566153301624, 7.07178820923151568476477608477, 8.788011870370973322928822144834, 9.420780712760980492787192496004, 10.67282492351914385288362718972, 11.19362867200670415681832096336, 12.88673692656949492280638062562

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