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Properties

Label 2-325-5.4-c5-0-7
Degree 22
Conductor 325325
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
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  − 2.89i·2-s − 3.45i·3-s + 23.6·4-s − 9.99·6-s + 148. i·7-s − 160. i·8-s + 231.·9-s − 712.·11-s − 81.7i·12-s + 169i·13-s + 428.·14-s + 290.·16-s + 1.13e3i·17-s − 668. i·18-s − 1.40e3·19-s + ⋯
L(s)  = 1  − 0.511i·2-s − 0.221i·3-s + 0.738·4-s − 0.113·6-s + 1.14i·7-s − 0.888i·8-s + 0.950·9-s − 1.77·11-s − 0.163i·12-s + 0.277i·13-s + 0.584·14-s + 0.284·16-s + 0.949i·17-s − 0.486i·18-s − 0.889·19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ325(274,⋅)\chi_{325} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :5/2), −0.447−0.894i)(2,\ 325,\ (\ :5/2),\ -0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 0.73591250280.7359125028
L(12)L(\frac12) ≈\approx 0.73591250280.7359125028
L(72)L(\frac{7}{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 1
13 1−169iT 1 - 169iT
good2 1+2.89iT−32T2 1 + 2.89iT - 32T^{2}
3 1+3.45iT−243T2 1 + 3.45iT - 243T^{2}
7 1−148.iT−1.68e4T2 1 - 148. iT - 1.68e4T^{2}
11 1+712.T+1.61e5T2 1 + 712.T + 1.61e5T^{2}
17 1−1.13e3iT−1.41e6T2 1 - 1.13e3iT - 1.41e6T^{2}
19 1+1.40e3T+2.47e6T2 1 + 1.40e3T + 2.47e6T^{2}
23 1−897.iT−6.43e6T2 1 - 897. iT - 6.43e6T^{2}
29 1+3.23e3T+2.05e7T2 1 + 3.23e3T + 2.05e7T^{2}
31 1+7.97e3T+2.86e7T2 1 + 7.97e3T + 2.86e7T^{2}
37 1+4.97e3iT−6.93e7T2 1 + 4.97e3iT - 6.93e7T^{2}
41 1+1.55e4T+1.15e8T2 1 + 1.55e4T + 1.15e8T^{2}
43 1+2.30e3iT−1.47e8T2 1 + 2.30e3iT - 1.47e8T^{2}
47 1−7.60e3iT−2.29e8T2 1 - 7.60e3iT - 2.29e8T^{2}
53 1+1.41e4iT−4.18e8T2 1 + 1.41e4iT - 4.18e8T^{2}
59 1+4.98e4T+7.14e8T2 1 + 4.98e4T + 7.14e8T^{2}
61 1+2.51e3T+8.44e8T2 1 + 2.51e3T + 8.44e8T^{2}
67 1−3.85e4iT−1.35e9T2 1 - 3.85e4iT - 1.35e9T^{2}
71 1−6.80e4T+1.80e9T2 1 - 6.80e4T + 1.80e9T^{2}
73 1+2.13e4iT−2.07e9T2 1 + 2.13e4iT - 2.07e9T^{2}
79 1+3.98e3T+3.07e9T2 1 + 3.98e3T + 3.07e9T^{2}
83 1+1.38e4iT−3.93e9T2 1 + 1.38e4iT - 3.93e9T^{2}
89 1+8.92e4T+5.58e9T2 1 + 8.92e4T + 5.58e9T^{2}
97 1−1.47e5iT−8.58e9T2 1 - 1.47e5iT - 8.58e9T^{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

−10.94307231824655215935111506083, −10.42090738740085153348332543837, −9.368542857421231667108198194735, −8.154552571186677808607664819490, −7.29704627487840507181404034676, −6.20029091804524294991739569841, −5.22660032675330831977259968577, −3.69371101474690230104215724911, −2.37580094412295084864019564854, −1.74202425904293740801679326167, 0.16306908020263547960215966687, 1.83509203596215331450287927992, 3.15603081753176675287243664753, 4.57051091219701523062830980597, 5.51178108220517065197003432970, 6.88779430140820356208103648068, 7.42301633187027906749652171299, 8.222496831202450606822491048662, 9.761041995205028206277629860463, 10.64277100467776110643572678749

Graph of the ZZ-function along the critical line