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Properties

Label 2-116-116.75-c1-0-0
Degree $2$
Conductor $116$
Sign $-0.189 - 0.981i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + (−1 + i)2-s + (−1.87 − 1.87i)3-s − 2i·4-s + 3i·5-s + 3.74·6-s + 3.74i·7-s + (2 + 2i)8-s + 4i·9-s + (−3 − 3i)10-s + (1.87 + 1.87i)11-s + (−3.74 + 3.74i)12-s − i·13-s + (−3.74 − 3.74i)14-s + (5.61 − 5.61i)15-s − 4·16-s + (−2 + 2i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.08 − 1.08i)3-s − i·4-s + 1.34i·5-s + 1.52·6-s + 1.41i·7-s + (0.707 + 0.707i)8-s + 1.33i·9-s + (−0.948 − 0.948i)10-s + (0.564 + 0.564i)11-s + (−1.08 + 1.08i)12-s − 0.277i·13-s + (−0.999 − 0.999i)14-s + (1.44 − 1.44i)15-s − 16-s + (−0.485 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ -0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298498 + 0.361469i\)
\(L(\frac12)\) \(\approx\) \(0.298498 + 0.361469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
29 \( 1 + (-5 + 2i)T \)
good3 \( 1 + (1.87 + 1.87i)T + 3iT^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 3.74iT - 7T^{2} \)
11 \( 1 + (-1.87 - 1.87i)T + 11iT^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 + (3.74 + 3.74i)T + 19iT^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
31 \( 1 + (-1.87 - 1.87i)T + 31iT^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + (1.87 + 1.87i)T + 43iT^{2} \)
47 \( 1 + (-5.61 + 5.61i)T - 47iT^{2} \)
53 \( 1 + 11T + 53T^{2} \)
59 \( 1 + 7.48iT - 59T^{2} \)
61 \( 1 + (-6 + 6i)T - 61iT^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 - 3.74T + 71T^{2} \)
73 \( 1 + (-4 - 4i)T + 73iT^{2} \)
79 \( 1 + (-5.61 - 5.61i)T + 79iT^{2} \)
83 \( 1 - 7.48iT - 83T^{2} \)
89 \( 1 + (3 - 3i)T - 89iT^{2} \)
97 \( 1 + (5 + 5i)T + 97iT^{2} \)
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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

−13.99041965933730714422576362333, −12.65706879043717683642159970045, −11.56060828463835413625584527916, −10.90932417052633920826340445174, −9.571492692656123471127217986950, −8.193014465733092064653586583382, −6.85677608123611615907340693801, −6.45967852627164154896448536540, −5.36684978221819068219376238158, −2.14238030262713219115281293140, 0.75583287941365810485845057426, 4.07316976052582075648399171318, 4.64171408069390877883636652319, 6.51549737799912795580443898492, 8.254528334466330180692993484889, 9.268277082029030067748900216553, 10.30410022348651632980331775905, 10.93833758581453565987264917908, 11.96857789871446007394828578027, 12.86397537255890729517450309117

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