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Properties

Label 2-115-115.22-c1-0-5
Degree $2$
Conductor $115$
Sign $-0.00202 + 0.999i$
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  + (−0.459 − 0.459i)2-s + (−1.79 + 1.79i)3-s − 1.57i·4-s + (−0.735 − 2.11i)5-s + 1.64·6-s + (2.52 − 2.52i)7-s + (−1.64 + 1.64i)8-s − 3.43i·9-s + (−0.631 + 1.30i)10-s − 4.72i·11-s + (2.83 + 2.83i)12-s + (0.0648 − 0.0648i)13-s − 2.32·14-s + (5.10 + 2.46i)15-s − 1.64·16-s + (−4.37 + 4.37i)17-s + ⋯
L(s)  = 1  + (−0.324 − 0.324i)2-s + (−1.03 + 1.03i)3-s − 0.789i·4-s + (−0.328 − 0.944i)5-s + 0.672·6-s + (0.955 − 0.955i)7-s + (−0.580 + 0.580i)8-s − 1.14i·9-s + (−0.199 + 0.413i)10-s − 1.42i·11-s + (0.817 + 0.817i)12-s + (0.0179 − 0.0179i)13-s − 0.620·14-s + (1.31 + 0.637i)15-s − 0.411·16-s + (−1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00202 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00202 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.00202 + 0.999i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ -0.00202 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420520 - 0.421372i\)
\(L(\frac12)\) \(\approx\) \(0.420520 - 0.421372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.735 + 2.11i)T \)
23 \( 1 + (-3.72 + 3.01i)T \)
good2 \( 1 + (0.459 + 0.459i)T + 2iT^{2} \)
3 \( 1 + (1.79 - 1.79i)T - 3iT^{2} \)
7 \( 1 + (-2.52 + 2.52i)T - 7iT^{2} \)
11 \( 1 + 4.72iT - 11T^{2} \)
13 \( 1 + (-0.0648 + 0.0648i)T - 13iT^{2} \)
17 \( 1 + (4.37 - 4.37i)T - 17iT^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
29 \( 1 - 3.20iT - 29T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + (-4.93 + 4.93i)T - 37iT^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 + (-0.563 - 0.563i)T + 43iT^{2} \)
47 \( 1 + (-2.59 - 2.59i)T + 47iT^{2} \)
53 \( 1 + (6.28 + 6.28i)T + 53iT^{2} \)
59 \( 1 - 5.31iT - 59T^{2} \)
61 \( 1 - 5.90iT - 61T^{2} \)
67 \( 1 + (0.206 - 0.206i)T - 67iT^{2} \)
71 \( 1 - 3.84T + 71T^{2} \)
73 \( 1 + (-2.10 + 2.10i)T - 73iT^{2} \)
79 \( 1 - 9.58T + 79T^{2} \)
83 \( 1 + (4.70 + 4.70i)T + 83iT^{2} \)
89 \( 1 - 8.50T + 89T^{2} \)
97 \( 1 + (2.58 - 2.58i)T - 97iT^{2} \)
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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

−13.31260717163698671701739278233, −11.72140139144194374755673250198, −10.94150287024324080713730072100, −10.59940588647375602316067187302, −9.217383423265220946034907952529, −8.241799495209998128844147032591, −6.16747856646806431705568190225, −5.07706666292556989717998265070, −4.22734240079532765004773847186, −0.869016560018669358046344868647, 2.44130488794554820901559658068, 4.78754416986342198516444215647, 6.41587364036169252486190802321, 7.21163256619227460387029832145, 7.977336736965331212469790641298, 9.470634841953509869952466629433, 11.26230481400729350601150141987, 11.74171518774315799132752805079, 12.46777869458588378930151836047, 13.63821142545128719409129064235

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