Properties

Label 2-115-23.16-c1-0-6
Degree $2$
Conductor $115$
Sign $0.358 + 0.933i$
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  + (1.29 − 1.49i)2-s + (0.749 − 0.481i)3-s + (−0.274 − 1.90i)4-s + (0.415 + 0.909i)5-s + (0.251 − 1.74i)6-s + (−3.44 + 1.01i)7-s + (0.118 + 0.0762i)8-s + (−0.916 + 2.00i)9-s + (1.90 + 0.558i)10-s + (−2.66 − 3.07i)11-s + (−1.12 − 1.29i)12-s + (0.453 + 0.133i)13-s + (−2.95 + 6.47i)14-s + (0.749 + 0.481i)15-s + (3.96 − 1.16i)16-s + (−0.293 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.917 − 1.05i)2-s + (0.432 − 0.278i)3-s + (−0.137 − 0.954i)4-s + (0.185 + 0.406i)5-s + (0.102 − 0.713i)6-s + (−1.30 + 0.382i)7-s + (0.0419 + 0.0269i)8-s + (−0.305 + 0.668i)9-s + (0.601 + 0.176i)10-s + (−0.803 − 0.927i)11-s + (−0.324 − 0.374i)12-s + (0.125 + 0.0369i)13-s + (−0.789 + 1.72i)14-s + (0.193 + 0.124i)15-s + (0.992 − 0.291i)16-s + (−0.0712 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35597 - 0.932119i\)
\(L(\frac12)\) \(\approx\) \(1.35597 - 0.932119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-3.46 - 3.31i)T \)
good2 \( 1 + (-1.29 + 1.49i)T + (-0.284 - 1.97i)T^{2} \)
3 \( 1 + (-0.749 + 0.481i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (3.44 - 1.01i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (2.66 + 3.07i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.453 - 0.133i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.293 - 2.04i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (1.10 + 7.67i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.637 + 4.43i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-4.14 - 2.66i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-0.140 + 0.306i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-1.21 - 2.65i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.28 + 1.46i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 9.09T + 47T^{2} \)
53 \( 1 + (4.31 - 1.26i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (14.0 + 4.12i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-3.64 - 2.34i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (0.224 - 0.259i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-6.18 + 7.13i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.367 + 2.55i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-16.5 - 4.86i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (1.07 - 2.35i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (7.82 - 5.02i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-6.27 - 13.7i)T + (-63.5 + 73.3i)T^{2} \)
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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

−13.43743892938195797420042335216, −12.63783779782513450026972865206, −11.27820191448396762555793155636, −10.64864480074470671950168689443, −9.339300173730871792172397044829, −7.988128856525370930784835678138, −6.40283891959870817880546888677, −5.09023698659922611471583381210, −3.23285776136797172714924761229, −2.57043539026161584299026106975, 3.30583104773865710796504159173, 4.56673263289293307876207558746, 5.93854869949751128617649106428, 6.85910561094829664224999392537, 8.107055321707819537651008151491, 9.531030626602611767428766151310, 10.30795272941466899738504120187, 12.41565092767705846723939342951, 12.88573463111070749241308169962, 13.94265902820235978077881382247

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