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Properties

Label 2-650-5.4-c5-0-35
Degree 22
Conductor 650650
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
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  − 4i·2-s + 9.04i·3-s − 16·4-s + 36.1·6-s + 77.0i·7-s + 64i·8-s + 161.·9-s + 463.·11-s − 144. i·12-s + 169i·13-s + 308.·14-s + 256·16-s + 1.86e3i·17-s − 644. i·18-s + 618.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.580i·3-s − 0.5·4-s + 0.410·6-s + 0.594i·7-s + 0.353i·8-s + 0.663·9-s + 1.15·11-s − 0.290i·12-s + 0.277i·13-s + 0.420·14-s + 0.250·16-s + 1.56i·17-s − 0.469i·18-s + 0.392·19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{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: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :5/2), 0.447−0.894i)(2,\ 650,\ (\ :5/2),\ 0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 2.3118542622.311854262
L(12)L(\frac12) ≈\approx 2.3118542622.311854262
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)
bad2 1+4iT 1 + 4iT
5 1 1
13 1−169iT 1 - 169iT
good3 1−9.04iT−243T2 1 - 9.04iT - 243T^{2}
7 1−77.0iT−1.68e4T2 1 - 77.0iT - 1.68e4T^{2}
11 1−463.T+1.61e5T2 1 - 463.T + 1.61e5T^{2}
17 1−1.86e3iT−1.41e6T2 1 - 1.86e3iT - 1.41e6T^{2}
19 1−618.T+2.47e6T2 1 - 618.T + 2.47e6T^{2}
23 1−1.71e3iT−6.43e6T2 1 - 1.71e3iT - 6.43e6T^{2}
29 1−5.86e3T+2.05e7T2 1 - 5.86e3T + 2.05e7T^{2}
31 1+545.T+2.86e7T2 1 + 545.T + 2.86e7T^{2}
37 1+1.23e4iT−6.93e7T2 1 + 1.23e4iT - 6.93e7T^{2}
41 1+1.77e4T+1.15e8T2 1 + 1.77e4T + 1.15e8T^{2}
43 1+1.26e4iT−1.47e8T2 1 + 1.26e4iT - 1.47e8T^{2}
47 1+1.59e4iT−2.29e8T2 1 + 1.59e4iT - 2.29e8T^{2}
53 1−2.79e4iT−4.18e8T2 1 - 2.79e4iT - 4.18e8T^{2}
59 1−2.22e4T+7.14e8T2 1 - 2.22e4T + 7.14e8T^{2}
61 1−5.54e3T+8.44e8T2 1 - 5.54e3T + 8.44e8T^{2}
67 1−5.96e4iT−1.35e9T2 1 - 5.96e4iT - 1.35e9T^{2}
71 1+6.72e4T+1.80e9T2 1 + 6.72e4T + 1.80e9T^{2}
73 1+6.73e4iT−2.07e9T2 1 + 6.73e4iT - 2.07e9T^{2}
79 1−4.90e4T+3.07e9T2 1 - 4.90e4T + 3.07e9T^{2}
83 1+1.25e4iT−3.93e9T2 1 + 1.25e4iT - 3.93e9T^{2}
89 1−1.36e3T+5.58e9T2 1 - 1.36e3T + 5.58e9T^{2}
97 1−5.80e4iT−8.58e9T2 1 - 5.80e4iT - 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

−10.11336037001943811248353186732, −9.080711470518403085407296672574, −8.671214185374675549537889170108, −7.31713844392224610055770436578, −6.23680823340364545875570562925, −5.20519040235512182770908392387, −4.09192726373308880786002949974, −3.56270454889092994946501765573, −2.06599223186808326222511489279, −1.16729531637881375391571567905, 0.57361469626341049233375630817, 1.37769705312942184495820928695, 3.01725868144286100281246537927, 4.27037542542211975411927471827, 5.03539546285517689201827733460, 6.50764220212980506986409170547, 6.82095483841854668086942861749, 7.69787123293199042388872038979, 8.568118250657419652031211602614, 9.621188136452250106392456576217

Graph of the ZZ-function along the critical line