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Properties

Label 2-115-5.4-c1-0-5
Degree $2$
Conductor $115$
Sign $0.447 + 0.894i$
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  − 2i·2-s + 2i·3-s − 2·4-s + (2 − i)5-s + 4·6-s − i·7-s − 9-s + (−2 − 4i)10-s − 4i·12-s + 2i·13-s − 2·14-s + (2 + 4i)15-s − 4·16-s + 5i·17-s + 2i·18-s − 8·19-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.15i·3-s − 4-s + (0.894 − 0.447i)5-s + 1.63·6-s − 0.377i·7-s − 0.333·9-s + (−0.632 − 1.26i)10-s − 1.15i·12-s + 0.554i·13-s − 0.534·14-s + (0.516 + 1.03i)15-s − 16-s + 1.21i·17-s + 0.471i·18-s − 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981071 - 0.606335i\)
\(L(\frac12)\) \(\approx\) \(0.981071 - 0.606335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 + i)T \)
23 \( 1 + iT \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 13T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−13.05246177038342160912014592331, −12.38983710253564654873592817997, −10.79793141301638198368689823549, −10.51457609821294924638293607291, −9.525399979740723914726144298645, −8.700159440524611374985708013508, −6.41592975336404706773088792020, −4.69585919052214494538236029718, −3.81238823938529052507809279733, −1.95925130624910070682789863149, 2.32076288399978477374016814777, 5.15554761100040160035616084601, 6.32012520794873606805595248351, 6.89015235144440048227525555321, 7.983831506816156597376228711386, 9.049433393000593893421739820103, 10.51132106280047819844563445919, 11.98903452670738403604747804450, 13.18232453524892023749560996127, 13.78101005206886419779092802530

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