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Properties

Label 2-115-23.22-c2-0-7
Degree $2$
Conductor $115$
Sign $0.765 - 0.643i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5.33·3-s − 3·4-s + 2.23i·5-s − 5.33·6-s + 13.3i·7-s + 7·8-s + 19.4·9-s − 2.23i·10-s − 9.13i·11-s − 16.0·12-s − 0.209·13-s − 13.3i·14-s + 11.9i·15-s + 5·16-s − 4.28i·17-s + ⋯
L(s)  = 1  − 0.5·2-s + 1.77·3-s − 0.750·4-s + 0.447i·5-s − 0.889·6-s + 1.90i·7-s + 0.875·8-s + 2.16·9-s − 0.223i·10-s − 0.830i·11-s − 1.33·12-s − 0.0161·13-s − 0.951i·14-s + 0.795i·15-s + 0.312·16-s − 0.251i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49262 + 0.543800i\)
\(L(\frac12)\) \(\approx\) \(1.49262 + 0.543800i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23iT \)
23 \( 1 + (-14.7 - 17.6i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 - 5.33T + 9T^{2} \)
7 \( 1 - 13.3iT - 49T^{2} \)
11 \( 1 + 9.13iT - 121T^{2} \)
13 \( 1 + 0.209T + 169T^{2} \)
17 \( 1 + 4.28iT - 289T^{2} \)
19 \( 1 + 26.7iT - 361T^{2} \)
29 \( 1 + 18.0T + 841T^{2} \)
31 \( 1 + 35.3T + 961T^{2} \)
37 \( 1 + 13.2iT - 1.36e3T^{2} \)
41 \( 1 - 11.5T + 1.68e3T^{2} \)
43 \( 1 + 48.4iT - 1.84e3T^{2} \)
47 \( 1 - 21.5T + 2.20e3T^{2} \)
53 \( 1 + 40.2iT - 2.80e3T^{2} \)
59 \( 1 - 105.T + 3.48e3T^{2} \)
61 \( 1 + 81.0iT - 3.72e3T^{2} \)
67 \( 1 - 58.3iT - 4.48e3T^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + 30.0T + 5.32e3T^{2} \)
79 \( 1 - 26.2iT - 6.24e3T^{2} \)
83 \( 1 + 14.7iT - 6.88e3T^{2} \)
89 \( 1 - 142. iT - 7.92e3T^{2} \)
97 \( 1 - 13.2iT - 9.40e3T^{2} \)
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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

−13.53966066988198753880645382753, −12.80536408291191480208209986019, −11.23496690014944208122291040454, −9.636891660100862269721165939645, −8.972652254668136294070248833253, −8.524307903000978642463587681570, −7.30007825790952817789194488208, −5.33270537542635484346789400061, −3.48268940153626403849298545514, −2.29803173703585481730042030668, 1.43157600375623608963538125783, 3.74235022587089554041871830871, 4.45830583387559563704159452509, 7.29891986670434825674430912014, 7.86259980938682995487821719723, 8.899267803103612992170145411614, 9.855260098341279080170007856603, 10.46368428562786659704501399588, 12.82590390188402175164606512608, 13.27425395120440431860633605111

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