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Properties

Label 2-115-115.68-c1-0-1
Degree $2$
Conductor $115$
Sign $-0.391 - 0.920i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.819 + 0.819i)2-s + (1.37 + 1.37i)3-s + 0.655i·4-s + (−2.23 − 0.0985i)5-s − 2.26·6-s + (1.78 + 1.78i)7-s + (−2.17 − 2.17i)8-s + 0.799i·9-s + (1.91 − 1.75i)10-s − 1.86i·11-s + (−0.903 + 0.903i)12-s + (1.52 + 1.52i)13-s − 2.92·14-s + (−2.94 − 3.21i)15-s + 2.26·16-s + (5.17 + 5.17i)17-s + ⋯
L(s)  = 1  + (−0.579 + 0.579i)2-s + (0.795 + 0.795i)3-s + 0.327i·4-s + (−0.999 − 0.0440i)5-s − 0.922·6-s + (0.674 + 0.674i)7-s + (−0.769 − 0.769i)8-s + 0.266i·9-s + (0.604 − 0.553i)10-s − 0.561i·11-s + (−0.260 + 0.260i)12-s + (0.422 + 0.422i)13-s − 0.782·14-s + (−0.759 − 0.830i)15-s + 0.565·16-s + (1.25 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494077 + 0.747417i\)
\(L(\frac12)\) \(\approx\) \(0.494077 + 0.747417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.0985i)T \)
23 \( 1 + (-2.59 + 4.03i)T \)
good2 \( 1 + (0.819 - 0.819i)T - 2iT^{2} \)
3 \( 1 + (-1.37 - 1.37i)T + 3iT^{2} \)
7 \( 1 + (-1.78 - 1.78i)T + 7iT^{2} \)
11 \( 1 + 1.86iT - 11T^{2} \)
13 \( 1 + (-1.52 - 1.52i)T + 13iT^{2} \)
17 \( 1 + (-5.17 - 5.17i)T + 17iT^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
29 \( 1 + 4.99iT - 29T^{2} \)
31 \( 1 - 0.833T + 31T^{2} \)
37 \( 1 + (3.00 + 3.00i)T + 37iT^{2} \)
41 \( 1 - 0.310T + 41T^{2} \)
43 \( 1 + (-2.17 + 2.17i)T - 43iT^{2} \)
47 \( 1 + (8.19 - 8.19i)T - 47iT^{2} \)
53 \( 1 + (6.21 - 6.21i)T - 53iT^{2} \)
59 \( 1 - 2.19iT - 59T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (-1.14 - 1.14i)T + 67iT^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + (6.62 + 6.62i)T + 73iT^{2} \)
79 \( 1 + 1.36T + 79T^{2} \)
83 \( 1 + (0.256 - 0.256i)T - 83iT^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-9.99 - 9.99i)T + 97iT^{2} \)
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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

−14.46424104119751841746832404004, −12.72272368367635035674581221557, −11.84514179321693892971599069305, −10.60980849307713490241887756017, −9.162295034814728956899744631447, −8.430200373353246129007722806752, −7.922803403882221048966732435732, −6.26753931354396119740034447979, −4.28973311649119010187481417539, −3.24606370444421455850381099673, 1.33969364083138672952590430689, 3.10788967710628497797005283568, 5.00278980586321467571912112410, 7.10191549608368057637486541135, 7.904495427746447232569996363120, 8.802124599936240829724146809311, 10.17610880362780821698484101732, 11.14604354914274675230755431497, 12.03449150598206984856302591904, 13.22637695840639230288759597938

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