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Properties

Label 2-115-23.6-c3-0-16
Degree 22
Conductor 115115
Sign −0.459+0.888i-0.459 + 0.888i
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.459+0.888i) Λ‾(4−s)\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}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((−0.459+0.888i) Λ‾(1−s)\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: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.459+0.888i-0.459 + 0.888i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(6,⋅)\chi_{115} (6, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), −0.459+0.888i)(2,\ 115,\ (\ :3/2),\ -0.459 + 0.888i)

Particular Values

L(2)L(2) ≈\approx 0.786705−1.29286i0.786705 - 1.29286i
L(12)L(\frac12) ≈\approx 0.786705−1.29286i0.786705 - 1.29286i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−4.20+2.70i)T 1 + (-4.20 + 2.70i)T
23 1+(65.8+88.4i)T 1 + (65.8 + 88.4i)T
good2 1+(−0.857+1.87i)T+(−5.23−6.04i)T2 1 + (-0.857 + 1.87i)T + (-5.23 - 6.04i)T^{2}
3 1+(4.53+1.33i)T+(22.7+14.5i)T2 1 + (4.53 + 1.33i)T + (22.7 + 14.5i)T^{2}
7 1+(−1.83+12.7i)T+(−329.−96.6i)T2 1 + (-1.83 + 12.7i)T + (-329. - 96.6i)T^{2}
11 1+(0.872+1.91i)T+(−871.+1.00e3i)T2 1 + (0.872 + 1.91i)T + (-871. + 1.00e3i)T^{2}
13 1+(8.93+62.1i)T+(−2.10e3+618.i)T2 1 + (8.93 + 62.1i)T + (-2.10e3 + 618. i)T^{2}
17 1+(−11.3+13.1i)T+(−699.−4.86e3i)T2 1 + (-11.3 + 13.1i)T + (-699. - 4.86e3i)T^{2}
19 1+(38.9+44.9i)T+(−976.+6.78e3i)T2 1 + (38.9 + 44.9i)T + (-976. + 6.78e3i)T^{2}
29 1+(−5.07+5.85i)T+(−3.47e3−2.41e4i)T2 1 + (-5.07 + 5.85i)T + (-3.47e3 - 2.41e4i)T^{2}
31 1+(−12.9+3.80i)T+(2.50e4−1.61e4i)T2 1 + (-12.9 + 3.80i)T + (2.50e4 - 1.61e4i)T^{2}
37 1+(−311.−199.i)T+(2.10e4+4.60e4i)T2 1 + (-311. - 199. i)T + (2.10e4 + 4.60e4i)T^{2}
41 1+(−33.4+21.4i)T+(2.86e4−6.26e4i)T2 1 + (-33.4 + 21.4i)T + (2.86e4 - 6.26e4i)T^{2}
43 1+(−213.−62.6i)T+(6.68e4+4.29e4i)T2 1 + (-213. - 62.6i)T + (6.68e4 + 4.29e4i)T^{2}
47 1+85.8T+1.03e5T2 1 + 85.8T + 1.03e5T^{2}
53 1+(−64.1+446.i)T+(−1.42e5−4.19e4i)T2 1 + (-64.1 + 446. i)T + (-1.42e5 - 4.19e4i)T^{2}
59 1+(−31.7−220.i)T+(−1.97e5+5.78e4i)T2 1 + (-31.7 - 220. i)T + (-1.97e5 + 5.78e4i)T^{2}
61 1+(405.−119.i)T+(1.90e5−1.22e5i)T2 1 + (405. - 119. i)T + (1.90e5 - 1.22e5i)T^{2}
67 1+(291.−639.i)T+(−1.96e5−2.27e5i)T2 1 + (291. - 639. i)T + (-1.96e5 - 2.27e5i)T^{2}
71 1+(416.−912.i)T+(−2.34e5−2.70e5i)T2 1 + (416. - 912. i)T + (-2.34e5 - 2.70e5i)T^{2}
73 1+(−328.−378.i)T+(−5.53e4+3.85e5i)T2 1 + (-328. - 378. i)T + (-5.53e4 + 3.85e5i)T^{2}
79 1+(−87.5−609.i)T+(−4.73e5+1.38e5i)T2 1 + (-87.5 - 609. i)T + (-4.73e5 + 1.38e5i)T^{2}
83 1+(859.+552.i)T+(2.37e5+5.20e5i)T2 1 + (859. + 552. i)T + (2.37e5 + 5.20e5i)T^{2}
89 1+(21.6+6.34i)T+(5.93e5+3.81e5i)T2 1 + (21.6 + 6.34i)T + (5.93e5 + 3.81e5i)T^{2}
97 1+(−1.34e3+861.i)T+(3.79e5−8.30e5i)T2 1 + (-1.34e3 + 861. i)T + (3.79e5 - 8.30e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line