Properties

Label 2-115-23.6-c3-0-16
Degree $2$
Conductor $115$
Sign $-0.459 + 0.888i$
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  + (0.857 − 1.87i)2-s + (−4.53 − 1.33i)3-s + (2.44 + 2.82i)4-s + (4.20 − 2.70i)5-s + (−6.38 + 7.37i)6-s + (1.83 − 12.7i)7-s + (23.2 − 6.82i)8-s + (−3.92 − 2.51i)9-s + (−1.46 − 10.2i)10-s + (−0.872 − 1.91i)11-s + (−7.34 − 16.0i)12-s + (−8.93 − 62.1i)13-s + (−22.4 − 14.4i)14-s + (−22.6 + 6.65i)15-s + (2.86 − 19.8i)16-s + (11.3 − 13.1i)17-s + ⋯
L(s)  = 1  + (0.303 − 0.663i)2-s + (−0.872 − 0.256i)3-s + (0.306 + 0.353i)4-s + (0.376 − 0.241i)5-s + (−0.434 + 0.501i)6-s + (0.0993 − 0.690i)7-s + (1.02 − 0.301i)8-s + (−0.145 − 0.0933i)9-s + (−0.0464 − 0.323i)10-s + (−0.0239 − 0.0523i)11-s + (−0.176 − 0.386i)12-s + (−0.190 − 1.32i)13-s + (−0.428 − 0.275i)14-s + (−0.390 + 0.114i)15-s + (0.0446 − 0.310i)16-s + (0.162 − 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.459 + 0.888i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ -0.459 + 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.786705 - 1.29286i\)
\(L(\frac12)\) \(\approx\) \(0.786705 - 1.29286i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.20 + 2.70i)T \)
23 \( 1 + (65.8 + 88.4i)T \)
good2 \( 1 + (-0.857 + 1.87i)T + (-5.23 - 6.04i)T^{2} \)
3 \( 1 + (4.53 + 1.33i)T + (22.7 + 14.5i)T^{2} \)
7 \( 1 + (-1.83 + 12.7i)T + (-329. - 96.6i)T^{2} \)
11 \( 1 + (0.872 + 1.91i)T + (-871. + 1.00e3i)T^{2} \)
13 \( 1 + (8.93 + 62.1i)T + (-2.10e3 + 618. i)T^{2} \)
17 \( 1 + (-11.3 + 13.1i)T + (-699. - 4.86e3i)T^{2} \)
19 \( 1 + (38.9 + 44.9i)T + (-976. + 6.78e3i)T^{2} \)
29 \( 1 + (-5.07 + 5.85i)T + (-3.47e3 - 2.41e4i)T^{2} \)
31 \( 1 + (-12.9 + 3.80i)T + (2.50e4 - 1.61e4i)T^{2} \)
37 \( 1 + (-311. - 199. i)T + (2.10e4 + 4.60e4i)T^{2} \)
41 \( 1 + (-33.4 + 21.4i)T + (2.86e4 - 6.26e4i)T^{2} \)
43 \( 1 + (-213. - 62.6i)T + (6.68e4 + 4.29e4i)T^{2} \)
47 \( 1 + 85.8T + 1.03e5T^{2} \)
53 \( 1 + (-64.1 + 446. i)T + (-1.42e5 - 4.19e4i)T^{2} \)
59 \( 1 + (-31.7 - 220. i)T + (-1.97e5 + 5.78e4i)T^{2} \)
61 \( 1 + (405. - 119. i)T + (1.90e5 - 1.22e5i)T^{2} \)
67 \( 1 + (291. - 639. i)T + (-1.96e5 - 2.27e5i)T^{2} \)
71 \( 1 + (416. - 912. i)T + (-2.34e5 - 2.70e5i)T^{2} \)
73 \( 1 + (-328. - 378. i)T + (-5.53e4 + 3.85e5i)T^{2} \)
79 \( 1 + (-87.5 - 609. i)T + (-4.73e5 + 1.38e5i)T^{2} \)
83 \( 1 + (859. + 552. i)T + (2.37e5 + 5.20e5i)T^{2} \)
89 \( 1 + (21.6 + 6.34i)T + (5.93e5 + 3.81e5i)T^{2} \)
97 \( 1 + (-1.34e3 + 861. i)T + (3.79e5 - 8.30e5i)T^{2} \)
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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

−12.68664436214891286589329215043, −11.73751036006451597848404885361, −10.85492177958315826558287326958, −10.07189869849107678466996453366, −8.280902612785023796776009618417, −7.07503898370274861905037227549, −5.84225181351286041536290272412, −4.44872545986565506922947571246, −2.79507685397497764525654113303, −0.835283235978681396396039980696, 2.04212560835106277607613261806, 4.54917602759371861178123682420, 5.75307528758562650194636613242, 6.29195164291043475728720090994, 7.67078561895492100186351908245, 9.256734621639999220820002253744, 10.46600160554191048727852670362, 11.32592290414641121413228550954, 12.20363278908258299208735129835, 13.76336875756165538654308603507

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