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Properties

Label 2-350-5.4-c5-0-5
Degree 22
Conductor 350350
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 56.134356.1343
Root an. cond. 7.492287.49228
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 − 30.5i·3-s − 16·4-s + 122.·6-s − 49i·7-s − 64i·8-s − 688.·9-s + 392.·11-s + 488. i·12-s + 631. i·13-s + 196·14-s + 256·16-s − 1.37e3i·17-s − 2.75e3i·18-s − 1.49e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.95i·3-s − 0.5·4-s + 1.38·6-s − 0.377i·7-s − 0.353i·8-s − 2.83·9-s + 0.977·11-s + 0.978i·12-s + 1.03i·13-s + 0.267·14-s + 0.250·16-s − 1.15i·17-s − 2.00i·18-s − 0.951·19-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+5/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{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: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 56.134356.1343
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ350(99,⋅)\chi_{350} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :5/2), 0.447−0.894i)(2,\ 350,\ (\ :5/2),\ 0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 0.80258661940.8025866194
L(12)L(\frac12) ≈\approx 0.80258661940.8025866194
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
7 1+49iT 1 + 49iT
good3 1+30.5iT−243T2 1 + 30.5iT - 243T^{2}
11 1−392.T+1.61e5T2 1 - 392.T + 1.61e5T^{2}
13 1−631.iT−3.71e5T2 1 - 631. iT - 3.71e5T^{2}
17 1+1.37e3iT−1.41e6T2 1 + 1.37e3iT - 1.41e6T^{2}
19 1+1.49e3T+2.47e6T2 1 + 1.49e3T + 2.47e6T^{2}
23 1−4.57e3iT−6.43e6T2 1 - 4.57e3iT - 6.43e6T^{2}
29 1+2.70e3T+2.05e7T2 1 + 2.70e3T + 2.05e7T^{2}
31 1+6.93e3T+2.86e7T2 1 + 6.93e3T + 2.86e7T^{2}
37 1−1.47e3iT−6.93e7T2 1 - 1.47e3iT - 6.93e7T^{2}
41 1−1.47e3T+1.15e8T2 1 - 1.47e3T + 1.15e8T^{2}
43 1−1.07e4iT−1.47e8T2 1 - 1.07e4iT - 1.47e8T^{2}
47 1+6.47e3iT−2.29e8T2 1 + 6.47e3iT - 2.29e8T^{2}
53 1+3.26e3iT−4.18e8T2 1 + 3.26e3iT - 4.18e8T^{2}
59 1−2.92e4T+7.14e8T2 1 - 2.92e4T + 7.14e8T^{2}
61 1−3.64e4T+8.44e8T2 1 - 3.64e4T + 8.44e8T^{2}
67 1+828.iT−1.35e9T2 1 + 828. iT - 1.35e9T^{2}
71 1−2.80e4T+1.80e9T2 1 - 2.80e4T + 1.80e9T^{2}
73 1−7.61e4iT−2.07e9T2 1 - 7.61e4iT - 2.07e9T^{2}
79 1+1.07e4T+3.07e9T2 1 + 1.07e4T + 3.07e9T^{2}
83 1+9.40e4iT−3.93e9T2 1 + 9.40e4iT - 3.93e9T^{2}
89 1−4.35e4T+5.58e9T2 1 - 4.35e4T + 5.58e9T^{2}
97 1−3.41e4iT−8.58e9T2 1 - 3.41e4iT - 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

−11.36643742148251685439886779579, −9.440416966792007375264390636997, −8.731595940885389872115164538467, −7.60793472131199628171980277959, −7.04184223399755267392231652464, −6.39380842962294787202257320497, −5.34764847568626433534420757804, −3.69797883337391125816662354975, −2.05410001086205215029834011087, −1.05555474268206049894674869583, 0.22983239363143018143963558278, 2.33695878649934662924640133673, 3.58518100012768525807928308158, 4.18135769023654329170920729020, 5.27630323059252197348574169161, 6.18847043161505100473618900970, 8.358675668637472495116638971210, 8.852656148892811130877915081459, 9.750316598884825227661974465408, 10.63038558236662043244008678136

Graph of the ZZ-function along the critical line