Properties

Label 2-115-5.4-c5-0-12
Degree 22
Conductor 115115
Sign 0.171+0.985i-0.171 + 0.985i
Analytic cond. 18.444118.4441
Root an. cond. 4.294664.29466
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.29i·2-s + 20.3i·3-s − 36.8·4-s + (55.0 + 9.59i)5-s − 168.·6-s + 101. i·7-s − 39.8i·8-s − 169.·9-s + (−79.5 + 456. i)10-s − 362.·11-s − 747. i·12-s − 540. i·13-s − 845.·14-s + (−194. + 1.11e3i)15-s − 847.·16-s + 561. i·17-s + ⋯
L(s)  = 1  + 1.46i·2-s + 1.30i·3-s − 1.15·4-s + (0.985 + 0.171i)5-s − 1.91·6-s + 0.786i·7-s − 0.219i·8-s − 0.698·9-s + (−0.251 + 1.44i)10-s − 0.903·11-s − 1.49i·12-s − 0.886i·13-s − 1.15·14-s + (−0.223 + 1.28i)15-s − 0.827·16-s + 0.471i·17-s + ⋯

Functional equation

Λ(s)=(115s/2ΓC(s)L(s)=((0.171+0.985i)Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(115s/2ΓC(s+5/2)L(s)=((0.171+0.985i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5235 \cdot 23
Sign: 0.171+0.985i-0.171 + 0.985i
Analytic conductor: 18.444118.4441
Root analytic conductor: 4.294664.29466
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ115(24,)\chi_{115} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :5/2), 0.171+0.985i)(2,\ 115,\ (\ :5/2),\ -0.171 + 0.985i)

Particular Values

L(3)L(3) \approx 1.7248510751.724851075
L(12)L(\frac12) \approx 1.7248510751.724851075
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(55.09.59i)T 1 + (-55.0 - 9.59i)T
23 1+529iT 1 + 529iT
good2 18.29iT32T2 1 - 8.29iT - 32T^{2}
3 120.3iT243T2 1 - 20.3iT - 243T^{2}
7 1101.iT1.68e4T2 1 - 101. iT - 1.68e4T^{2}
11 1+362.T+1.61e5T2 1 + 362.T + 1.61e5T^{2}
13 1+540.iT3.71e5T2 1 + 540. iT - 3.71e5T^{2}
17 1561.iT1.41e6T2 1 - 561. iT - 1.41e6T^{2}
19 1+1.11e3T+2.47e6T2 1 + 1.11e3T + 2.47e6T^{2}
29 17.26e3T+2.05e7T2 1 - 7.26e3T + 2.05e7T^{2}
31 1450.T+2.86e7T2 1 - 450.T + 2.86e7T^{2}
37 14.96e3iT6.93e7T2 1 - 4.96e3iT - 6.93e7T^{2}
41 17.55e3T+1.15e8T2 1 - 7.55e3T + 1.15e8T^{2}
43 11.60e4iT1.47e8T2 1 - 1.60e4iT - 1.47e8T^{2}
47 1+1.94e4iT2.29e8T2 1 + 1.94e4iT - 2.29e8T^{2}
53 15.35e3iT4.18e8T2 1 - 5.35e3iT - 4.18e8T^{2}
59 1+4.93e4T+7.14e8T2 1 + 4.93e4T + 7.14e8T^{2}
61 11.63e3T+8.44e8T2 1 - 1.63e3T + 8.44e8T^{2}
67 1+3.44e4iT1.35e9T2 1 + 3.44e4iT - 1.35e9T^{2}
71 1+5.39e3T+1.80e9T2 1 + 5.39e3T + 1.80e9T^{2}
73 15.24e4iT2.07e9T2 1 - 5.24e4iT - 2.07e9T^{2}
79 1+4.43e3T+3.07e9T2 1 + 4.43e3T + 3.07e9T^{2}
83 19.18e4iT3.93e9T2 1 - 9.18e4iT - 3.93e9T^{2}
89 18.87e4T+5.58e9T2 1 - 8.87e4T + 5.58e9T^{2}
97 15.56e4iT8.58e9T2 1 - 5.56e4iT - 8.58e9T^{2}
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   L(s)=p j=12(1αj,pps)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

−13.78967302429692143960492154024, −12.67172748455404224041258252228, −10.81879547056388132248357173632, −10.04942344640460667265339055995, −8.964950815110959858449691136681, −8.074439358485842620303104597548, −6.42108048482741060673798472673, −5.52656829008007783805803883415, −4.72517758308867770575037365780, −2.70189249286172917619152542473, 0.62931305016555509234678932482, 1.71914438263393115023555760907, 2.66246861983355665848731167472, 4.53480990403258999517752986462, 6.32691510820317620921661145660, 7.38218327996330290651196602103, 8.896503689572419184503153731089, 10.08197249101645712285869536207, 10.85161134779635722526557843428, 12.09142181119958421857956983013

Graph of the ZZ-function along the critical line