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Properties

Label 2-95-95.94-c10-0-53
Degree 22
Conductor 9595
Sign 0.632+0.774i0.632 + 0.774i
Analytic cond. 60.358960.3589
Root an. cond. 7.769107.76910
Motivic weight 1010
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.02e3·4-s + (1.97e3 + 2.42e3i)5-s + 8.48e3i·7-s − 5.90e4·9-s − 2.03e5·11-s + 1.04e6·16-s − 1.85e6i·17-s − 2.47e6·19-s + (−2.02e6 − 2.47e6i)20-s + 1.18e7i·23-s + (−1.96e6 + 9.56e6i)25-s − 8.69e6i·28-s + (−2.05e7 + 1.67e7i)35-s + 6.04e7·36-s − 2.03e8i·43-s + 2.08e8·44-s + ⋯
L(s)  = 1  − 4-s + (0.632 + 0.774i)5-s + 0.504i·7-s − 0.999·9-s − 1.26·11-s + 16-s − 1.30i·17-s − 19-s + (−0.632 − 0.774i)20-s + 1.83i·23-s + (−0.200 + 0.979i)25-s − 0.504i·28-s + (−0.391 + 0.319i)35-s + 0.999·36-s − 1.38i·43-s + 1.26·44-s + ⋯

Functional equation

Λ(s)=(95s/2 ΓC(s) L(s)=((0.632+0.774i) Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(95s/2 ΓC(s+5) L(s)=((0.632+0.774i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9595    =    5⋅195 \cdot 19
Sign: 0.632+0.774i0.632 + 0.774i
Analytic conductor: 60.358960.3589
Root analytic conductor: 7.769107.76910
Motivic weight: 1010
Rational: no
Arithmetic: yes
Character: χ95(94,⋅)\chi_{95} (94, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 95, ( :5), 0.632+0.774i)(2,\ 95,\ (\ :5),\ 0.632 + 0.774i)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 0.612044−0.290556i0.612044 - 0.290556i
L(12)L(\frac12) ≈\approx 0.612044−0.290556i0.612044 - 0.290556i
L(6)L(6) 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+(−1.97e3−2.42e3i)T 1 + (-1.97e3 - 2.42e3i)T
19 1+2.47e6T 1 + 2.47e6T
good2 1+1.02e3T2 1 + 1.02e3T^{2}
3 1+5.90e4T2 1 + 5.90e4T^{2}
7 1−8.48e3iT−2.82e8T2 1 - 8.48e3iT - 2.82e8T^{2}
11 1+2.03e5T+2.59e10T2 1 + 2.03e5T + 2.59e10T^{2}
13 1+1.37e11T2 1 + 1.37e11T^{2}
17 1+1.85e6iT−2.01e12T2 1 + 1.85e6iT - 2.01e12T^{2}
23 1−1.18e7iT−4.14e13T2 1 - 1.18e7iT - 4.14e13T^{2}
29 1−4.20e14T2 1 - 4.20e14T^{2}
31 1−8.19e14T2 1 - 8.19e14T^{2}
37 1+4.80e15T2 1 + 4.80e15T^{2}
41 1−1.34e16T2 1 - 1.34e16T^{2}
43 1+2.03e8iT−2.16e16T2 1 + 2.03e8iT - 2.16e16T^{2}
47 1−4.28e7iT−5.25e16T2 1 - 4.28e7iT - 5.25e16T^{2}
53 1+1.74e17T2 1 + 1.74e17T^{2}
59 1−5.11e17T2 1 - 5.11e17T^{2}
61 1−1.60e9T+7.13e17T2 1 - 1.60e9T + 7.13e17T^{2}
67 1+1.82e18T2 1 + 1.82e18T^{2}
71 1−3.25e18T2 1 - 3.25e18T^{2}
73 1+2.70e9iT−4.29e18T2 1 + 2.70e9iT - 4.29e18T^{2}
79 1−9.46e18T2 1 - 9.46e18T^{2}
83 1+7.57e9iT−1.55e19T2 1 + 7.57e9iT - 1.55e19T^{2}
89 1−3.11e19T2 1 - 3.11e19T^{2}
97 1+7.37e19T2 1 + 7.37e19T^{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

−11.77719259641969960674549637429, −10.63396366213544073879169660943, −9.600166445566417064624825380321, −8.676299838494493369399258466472, −7.45176617241802238810016800197, −5.80047099553414431823568676264, −5.14378654070056368628474526800, −3.31919316396098611609328388716, −2.27361095777284880183677926131, −0.24820434471412867367521316336, 0.75020506806192037575935143114, 2.42135042513146881104795493235, 4.13208760491157723585707550606, 5.15730103531334606385314228817, 6.17217720397381422258065391455, 8.231373408226320965486263192293, 8.595756068064213234994245972359, 9.991202521767680121983389418604, 10.76794745697115588505466805548, 12.60068889380601118170171174584

Graph of the ZZ-function along the critical line