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Properties

Label 2-325-5.4-c5-0-47
Degree 22
Conductor 325325
Sign 0.447−0.894i0.447 - 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  + 0.396i·2-s + 11.4i·3-s + 31.8·4-s − 4.55·6-s + 8.04i·7-s + 25.3i·8-s + 110.·9-s + 335.·11-s + 366. i·12-s + 169i·13-s − 3.19·14-s + 1.00e3·16-s + 38.1i·17-s + 43.9i·18-s − 1.17e3·19-s + ⋯
L(s)  = 1  + 0.0701i·2-s + 0.737i·3-s + 0.995·4-s − 0.0517·6-s + 0.0620i·7-s + 0.139i·8-s + 0.456·9-s + 0.837·11-s + 0.733i·12-s + 0.277i·13-s − 0.00435·14-s + 0.985·16-s + 0.0320i·17-s + 0.0319i·18-s − 0.749·19-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: 0.447−0.894i0.447 - 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 3.1726821023.172682102
L(12)L(\frac12) ≈\approx 3.1726821023.172682102
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 1−0.396iT−32T2 1 - 0.396iT - 32T^{2}
3 1−11.4iT−243T2 1 - 11.4iT - 243T^{2}
7 1−8.04iT−1.68e4T2 1 - 8.04iT - 1.68e4T^{2}
11 1−335.T+1.61e5T2 1 - 335.T + 1.61e5T^{2}
17 1−38.1iT−1.41e6T2 1 - 38.1iT - 1.41e6T^{2}
19 1+1.17e3T+2.47e6T2 1 + 1.17e3T + 2.47e6T^{2}
23 1+1.67e3iT−6.43e6T2 1 + 1.67e3iT - 6.43e6T^{2}
29 1−6.70e3T+2.05e7T2 1 - 6.70e3T + 2.05e7T^{2}
31 1−7.85e3T+2.86e7T2 1 - 7.85e3T + 2.86e7T^{2}
37 1+1.71e3iT−6.93e7T2 1 + 1.71e3iT - 6.93e7T^{2}
41 1+1.00e4T+1.15e8T2 1 + 1.00e4T + 1.15e8T^{2}
43 1+1.08e4iT−1.47e8T2 1 + 1.08e4iT - 1.47e8T^{2}
47 1−1.29e4iT−2.29e8T2 1 - 1.29e4iT - 2.29e8T^{2}
53 1+2.63e3iT−4.18e8T2 1 + 2.63e3iT - 4.18e8T^{2}
59 1+4.95e3T+7.14e8T2 1 + 4.95e3T + 7.14e8T^{2}
61 1+4.37e4T+8.44e8T2 1 + 4.37e4T + 8.44e8T^{2}
67 1+9.17e3iT−1.35e9T2 1 + 9.17e3iT - 1.35e9T^{2}
71 1+1.30e4T+1.80e9T2 1 + 1.30e4T + 1.80e9T^{2}
73 1−4.17e4iT−2.07e9T2 1 - 4.17e4iT - 2.07e9T^{2}
79 1−4.98e4T+3.07e9T2 1 - 4.98e4T + 3.07e9T^{2}
83 1−8.61e4iT−3.93e9T2 1 - 8.61e4iT - 3.93e9T^{2}
89 1−6.72e4T+5.58e9T2 1 - 6.72e4T + 5.58e9T^{2}
97 1−1.75e5iT−8.58e9T2 1 - 1.75e5iT - 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

−10.72571184267423537906986321940, −10.24081113373247996810230521309, −9.154831279245133542781750289175, −8.130563526767445718772365978761, −6.86467150418391950247285975724, −6.28330625443407165857755591769, −4.82441467496599727968044140728, −3.84479867929998221671243409071, −2.53690896606775342189793642960, −1.21033263059149027270926338228, 0.940284335217437725910994376327, 1.86540403610743283646628988876, 3.09650945543894475618184885924, 4.49622016377348716970923409345, 6.12765866893591834816283564863, 6.69808431864783242605252017753, 7.57843028752942039430398243580, 8.512710830931869934540923669104, 9.896543809198597510729745232787, 10.62791589917596542648521873950

Graph of the ZZ-function along the critical line