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Properties

Label 2-325-5.4-c5-0-87
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  − 3.20i·2-s − 29.5i·3-s + 21.7·4-s − 94.8·6-s + 11.5i·7-s − 172. i·8-s − 632.·9-s − 596.·11-s − 642. i·12-s + 169i·13-s + 37.1·14-s + 142.·16-s − 2.09e3i·17-s + 2.02e3i·18-s − 35.4·19-s + ⋯
L(s)  = 1  − 0.566i·2-s − 1.89i·3-s + 0.678·4-s − 1.07·6-s + 0.0892i·7-s − 0.951i·8-s − 2.60·9-s − 1.48·11-s − 1.28i·12-s + 0.277i·13-s + 0.0506·14-s + 0.139·16-s − 1.75i·17-s + 1.47i·18-s − 0.0225·19-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 0.76592853570.7659285357
L(12)L(\frac12) ≈\approx 0.76592853570.7659285357
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+3.20iT−32T2 1 + 3.20iT - 32T^{2}
3 1+29.5iT−243T2 1 + 29.5iT - 243T^{2}
7 1−11.5iT−1.68e4T2 1 - 11.5iT - 1.68e4T^{2}
11 1+596.T+1.61e5T2 1 + 596.T + 1.61e5T^{2}
17 1+2.09e3iT−1.41e6T2 1 + 2.09e3iT - 1.41e6T^{2}
19 1+35.4T+2.47e6T2 1 + 35.4T + 2.47e6T^{2}
23 1−2.78e3iT−6.43e6T2 1 - 2.78e3iT - 6.43e6T^{2}
29 1−370.T+2.05e7T2 1 - 370.T + 2.05e7T^{2}
31 1−5.05e3T+2.86e7T2 1 - 5.05e3T + 2.86e7T^{2}
37 1+4.12e3iT−6.93e7T2 1 + 4.12e3iT - 6.93e7T^{2}
41 1+1.81e4T+1.15e8T2 1 + 1.81e4T + 1.15e8T^{2}
43 1+7.90e3iT−1.47e8T2 1 + 7.90e3iT - 1.47e8T^{2}
47 1−1.31e4iT−2.29e8T2 1 - 1.31e4iT - 2.29e8T^{2}
53 1−3.82e4iT−4.18e8T2 1 - 3.82e4iT - 4.18e8T^{2}
59 1+1.78e4T+7.14e8T2 1 + 1.78e4T + 7.14e8T^{2}
61 1−7.39e3T+8.44e8T2 1 - 7.39e3T + 8.44e8T^{2}
67 1−2.33e4iT−1.35e9T2 1 - 2.33e4iT - 1.35e9T^{2}
71 1+3.31e4T+1.80e9T2 1 + 3.31e4T + 1.80e9T^{2}
73 1+1.08e4iT−2.07e9T2 1 + 1.08e4iT - 2.07e9T^{2}
79 1−1.77e4T+3.07e9T2 1 - 1.77e4T + 3.07e9T^{2}
83 1+8.42e4iT−3.93e9T2 1 + 8.42e4iT - 3.93e9T^{2}
89 1−4.64e4T+5.58e9T2 1 - 4.64e4T + 5.58e9T^{2}
97 1+1.54e5iT−8.58e9T2 1 + 1.54e5iT - 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

−10.26464985074492341994357727996, −8.931060139785972302820022867597, −7.59707640601827919319344262269, −7.36955737541825208652762611246, −6.28396249907723201603327899088, −5.27461904287277902549120057686, −2.98992942803779082318067691672, −2.41218186200037421504477115783, −1.28991080725586359721882999750, −0.18475467836405768476338680412, 2.43072010846673202855284819410, 3.47725811711752918898264119339, 4.75152295609782881031287881609, 5.51719684595395905630295848273, 6.46913417080508007443984357826, 8.162413412389258628832969670734, 8.453174926825239788858410364217, 10.06799090164732486957934654880, 10.43600209282465574285677905962, 11.10012890585024235559798136618

Graph of the ZZ-function along the critical line