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Properties

Label 2-115-23.22-c6-0-44
Degree $2$
Conductor $115$
Sign $-0.991 + 0.132i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2·2-s + 36.0·3-s + 41.7·4-s − 55.9i·5-s − 370.·6-s − 481. i·7-s + 228.·8-s + 567.·9-s + 574. i·10-s − 1.21e3i·11-s + 1.50e3·12-s − 3.40e3·13-s + 4.95e3i·14-s − 2.01e3i·15-s − 5.02e3·16-s − 920. i·17-s + ⋯
L(s)  = 1  − 1.28·2-s + 1.33·3-s + 0.652·4-s − 0.447i·5-s − 1.71·6-s − 1.40i·7-s + 0.446·8-s + 0.778·9-s + 0.574i·10-s − 0.910i·11-s + 0.870·12-s − 1.55·13-s + 1.80i·14-s − 0.596i·15-s − 1.22·16-s − 0.187i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.991 + 0.132i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ -0.991 + 0.132i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.6449383696\)
\(L(\frac12)\) \(\approx\) \(0.6449383696\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 55.9iT \)
23 \( 1 + (-1.61e3 - 1.20e4i)T \)
good2 \( 1 + 10.2T + 64T^{2} \)
3 \( 1 - 36.0T + 729T^{2} \)
7 \( 1 + 481. iT - 1.17e5T^{2} \)
11 \( 1 + 1.21e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.40e3T + 4.82e6T^{2} \)
17 \( 1 + 920. iT - 2.41e7T^{2} \)
19 \( 1 - 2.61e3iT - 4.70e7T^{2} \)
29 \( 1 - 1.89e4T + 5.94e8T^{2} \)
31 \( 1 - 1.98e4T + 8.87e8T^{2} \)
37 \( 1 - 5.44e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.12e5T + 4.75e9T^{2} \)
43 \( 1 + 3.69e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.70e5T + 1.07e10T^{2} \)
53 \( 1 + 8.34e4iT - 2.21e10T^{2} \)
59 \( 1 - 2.15e4T + 4.21e10T^{2} \)
61 \( 1 - 9.15e4iT - 5.15e10T^{2} \)
67 \( 1 + 1.33e4iT - 9.04e10T^{2} \)
71 \( 1 + 4.64e5T + 1.28e11T^{2} \)
73 \( 1 - 5.58e3T + 1.51e11T^{2} \)
79 \( 1 + 6.48e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.76e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 - 3.29e5iT - 8.32e11T^{2} \)
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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

−11.66488269397657020018013993281, −10.22637186841471013368261531301, −9.715595326184737740174752724224, −8.584240641163357060598704487204, −7.899288081183274269479897579001, −7.05241078723834882767344715514, −4.67159293588277958220906988029, −3.22186139227792990176473544214, −1.56822042187183902074365901377, −0.26292141049042441078400098473, 2.04557399350907491828300617991, 2.71305053951747574922717770380, 4.77977510982823105869001980109, 6.83753889608708333374499908859, 7.895249549600383038776033127096, 8.706409154731541072977771592160, 9.515457170465739870516700490550, 10.20499976683695087097877127684, 11.78757416417856990224524258228, 12.84514982290449620315839621289

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