Properties

Label 2-115-115.22-c3-0-11
Degree $2$
Conductor $115$
Sign $0.964 + 0.265i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 − 2.66i)2-s + (−2.23 + 2.23i)3-s + 6.20i·4-s + (7.41 − 8.37i)5-s + 11.9·6-s + (−10.9 + 10.9i)7-s + (−4.78 + 4.78i)8-s + 17.0i·9-s + (−42.0 + 2.55i)10-s + 18.9i·11-s + (−13.8 − 13.8i)12-s + (43.9 − 43.9i)13-s + 58.1·14-s + (2.14 + 35.2i)15-s + 75.1·16-s + (−38.7 + 38.7i)17-s + ⋯
L(s)  = 1  + (−0.942 − 0.942i)2-s + (−0.429 + 0.429i)3-s + 0.775i·4-s + (0.662 − 0.748i)5-s + 0.809·6-s + (−0.588 + 0.588i)7-s + (−0.211 + 0.211i)8-s + 0.630i·9-s + (−1.33 + 0.0808i)10-s + 0.519i·11-s + (−0.333 − 0.333i)12-s + (0.937 − 0.937i)13-s + 1.10·14-s + (0.0368 + 0.606i)15-s + 1.17·16-s + (−0.552 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.838400 - 0.113353i\)
\(L(\frac12)\) \(\approx\) \(0.838400 - 0.113353i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.41 + 8.37i)T \)
23 \( 1 + (-109. - 10.7i)T \)
good2 \( 1 + (2.66 + 2.66i)T + 8iT^{2} \)
3 \( 1 + (2.23 - 2.23i)T - 27iT^{2} \)
7 \( 1 + (10.9 - 10.9i)T - 343iT^{2} \)
11 \( 1 - 18.9iT - 1.33e3T^{2} \)
13 \( 1 + (-43.9 + 43.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (38.7 - 38.7i)T - 4.91e3iT^{2} \)
19 \( 1 - 123.T + 6.85e3T^{2} \)
29 \( 1 - 59.0iT - 2.43e4T^{2} \)
31 \( 1 - 342.T + 2.97e4T^{2} \)
37 \( 1 + (302. - 302. i)T - 5.06e4iT^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + (-155. - 155. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-234. - 234. i)T + 1.03e5iT^{2} \)
53 \( 1 + (342. + 342. i)T + 1.48e5iT^{2} \)
59 \( 1 + 87.0iT - 2.05e5T^{2} \)
61 \( 1 - 565. iT - 2.26e5T^{2} \)
67 \( 1 + (4.05 - 4.05i)T - 3.00e5iT^{2} \)
71 \( 1 + 556.T + 3.57e5T^{2} \)
73 \( 1 + (-305. + 305. i)T - 3.89e5iT^{2} \)
79 \( 1 + 328.T + 4.93e5T^{2} \)
83 \( 1 + (-810. - 810. i)T + 5.71e5iT^{2} \)
89 \( 1 - 194.T + 7.04e5T^{2} \)
97 \( 1 + (-1.06e3 + 1.06e3i)T - 9.12e5iT^{2} \)
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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

−12.76694572204519277056677097313, −11.79168548375939974287764730716, −10.68413600824944339899874651847, −9.957477479024622232147344056761, −9.101402500252619188176040266422, −8.134854281233452000667373189363, −6.03144253019499606059641003578, −5.01064957684409569082091759515, −2.85106893985799599131786187832, −1.20408564579080471144051939777, 0.809614437334806712078780371344, 3.38682930705671789484534951679, 5.87061365833079225460053348521, 6.69220138097349106110669626590, 7.25950380572259024031261419603, 8.922962920189886156306030040997, 9.627496187491572016940478148753, 10.84318430532910649784498269481, 11.97323586794373419543073810174, 13.42774116446281368715591862468

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