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Properties

Label 2-325-5.4-c5-0-65
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  − 11.2i·2-s + 19.1i·3-s − 94.0·4-s + 215.·6-s − 70.9i·7-s + 696. i·8-s − 125.·9-s − 161.·11-s − 1.80e3i·12-s + 169i·13-s − 796.·14-s + 4.80e3·16-s + 121. i·17-s + 1.40e3i·18-s + 3.11e3·19-s + ⋯
L(s)  = 1  − 1.98i·2-s + 1.23i·3-s − 2.93·4-s + 2.44·6-s − 0.547i·7-s + 3.84i·8-s − 0.515·9-s − 0.401·11-s − 3.61i·12-s + 0.277i·13-s − 1.08·14-s + 4.69·16-s + 0.101i·17-s + 1.02i·18-s + 1.97·19-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 0.23196956030.2319695603
L(12)L(\frac12) ≈\approx 0.23196956030.2319695603
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+11.2iT−32T2 1 + 11.2iT - 32T^{2}
3 1−19.1iT−243T2 1 - 19.1iT - 243T^{2}
7 1+70.9iT−1.68e4T2 1 + 70.9iT - 1.68e4T^{2}
11 1+161.T+1.61e5T2 1 + 161.T + 1.61e5T^{2}
17 1−121.iT−1.41e6T2 1 - 121. iT - 1.41e6T^{2}
19 1−3.11e3T+2.47e6T2 1 - 3.11e3T + 2.47e6T^{2}
23 1+3.09e3iT−6.43e6T2 1 + 3.09e3iT - 6.43e6T^{2}
29 1+3.17e3T+2.05e7T2 1 + 3.17e3T + 2.05e7T^{2}
31 1+3.51e3T+2.86e7T2 1 + 3.51e3T + 2.86e7T^{2}
37 1+6.99e3iT−6.93e7T2 1 + 6.99e3iT - 6.93e7T^{2}
41 1+1.95e4T+1.15e8T2 1 + 1.95e4T + 1.15e8T^{2}
43 1−1.33e4iT−1.47e8T2 1 - 1.33e4iT - 1.47e8T^{2}
47 1+7.63e3iT−2.29e8T2 1 + 7.63e3iT - 2.29e8T^{2}
53 1−2.77e4iT−4.18e8T2 1 - 2.77e4iT - 4.18e8T^{2}
59 1+3.30e4T+7.14e8T2 1 + 3.30e4T + 7.14e8T^{2}
61 1+3.31e4T+8.44e8T2 1 + 3.31e4T + 8.44e8T^{2}
67 1+2.45e4iT−1.35e9T2 1 + 2.45e4iT - 1.35e9T^{2}
71 1−1.66e4T+1.80e9T2 1 - 1.66e4T + 1.80e9T^{2}
73 1+5.25e3iT−2.07e9T2 1 + 5.25e3iT - 2.07e9T^{2}
79 1−3.12e3T+3.07e9T2 1 - 3.12e3T + 3.07e9T^{2}
83 1+2.51e4iT−3.93e9T2 1 + 2.51e4iT - 3.93e9T^{2}
89 1+634.T+5.58e9T2 1 + 634.T + 5.58e9T^{2}
97 1+1.56e5iT−8.58e9T2 1 + 1.56e5iT - 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

−10.37495893611789225998182538609, −9.550810987197325993177104739603, −8.988462148402855868857561770484, −7.69511333762359243813683008810, −5.40658069393873363384542486104, −4.61576196491944571205065296332, −3.74909990046918467073142160851, −2.92836787524459980163867653520, −1.43476455682323609521995315516, −0.06993881135606415513468755086, 1.27054786711173400915979774472, 3.43806397749804201964257203973, 5.15779066931417445457433534902, 5.69245967362803549093878154765, 6.80735299296492054612977253762, 7.48204204050828274333532016217, 8.056722694638745836980227475367, 9.110731259413024074958497301342, 9.941441555435790639911985264019, 11.83221017044450593585265106091

Graph of the ZZ-function along the critical line