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Properties

Label 2-350-5.4-c5-0-33
Degree 22
Conductor 350350
Sign −0.447+0.894i-0.447 + 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 + 9i·3-s − 16·4-s + 36·6-s − 49i·7-s + 64i·8-s + 162·9-s − 187·11-s − 144i·12-s − 627i·13-s − 196·14-s + 256·16-s + 1.81e3i·17-s − 648i·18-s − 258·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s + 0.666·9-s − 0.465·11-s − 0.288i·12-s − 1.02i·13-s − 0.267·14-s + 0.250·16-s + 1.52i·17-s − 0.471i·18-s − 0.163·19-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+0.894i-0.447 + 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 1.4202452541.420245254
L(12)L(\frac12) ≈\approx 1.4202452541.420245254
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 1−9iT−243T2 1 - 9iT - 243T^{2}
11 1+187T+1.61e5T2 1 + 187T + 1.61e5T^{2}
13 1+627iT−3.71e5T2 1 + 627iT - 3.71e5T^{2}
17 1−1.81e3iT−1.41e6T2 1 - 1.81e3iT - 1.41e6T^{2}
19 1+258T+2.47e6T2 1 + 258T + 2.47e6T^{2}
23 1+2.97e3iT−6.43e6T2 1 + 2.97e3iT - 6.43e6T^{2}
29 1+1.29e3T+2.05e7T2 1 + 1.29e3T + 2.05e7T^{2}
31 1−1.91e3T+2.86e7T2 1 - 1.91e3T + 2.86e7T^{2}
37 1−6.57e3iT−6.93e7T2 1 - 6.57e3iT - 6.93e7T^{2}
41 1−6.67e3T+1.15e8T2 1 - 6.67e3T + 1.15e8T^{2}
43 1+3.17e3iT−1.47e8T2 1 + 3.17e3iT - 1.47e8T^{2}
47 1+2.20e4iT−2.29e8T2 1 + 2.20e4iT - 2.29e8T^{2}
53 1+2.61e4iT−4.18e8T2 1 + 2.61e4iT - 4.18e8T^{2}
59 1+3.93e3T+7.14e8T2 1 + 3.93e3T + 7.14e8T^{2}
61 1+4.87e4T+8.44e8T2 1 + 4.87e4T + 8.44e8T^{2}
67 1+4.48e4iT−1.35e9T2 1 + 4.48e4iT - 1.35e9T^{2}
71 1−6.37e4T+1.80e9T2 1 - 6.37e4T + 1.80e9T^{2}
73 1+6.04e4iT−2.07e9T2 1 + 6.04e4iT - 2.07e9T^{2}
79 1−4.37e4T+3.07e9T2 1 - 4.37e4T + 3.07e9T^{2}
83 1+9.72e4iT−3.93e9T2 1 + 9.72e4iT - 3.93e9T^{2}
89 1+4.55e4T+5.58e9T2 1 + 4.55e4T + 5.58e9T^{2}
97 1+5.72e4iT−8.58e9T2 1 + 5.72e4iT - 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

−10.47906732428098559445644187128, −9.800798676645035956057119740927, −8.586251534171562978272507600477, −7.77568503572756408848118064551, −6.39280228010322361253858911285, −5.11567945929523758931851025246, −4.17578634295150094482908847181, −3.23190278568680553878919085783, −1.79968063732603726623319593322, −0.40158329257998323542529266675, 1.14858579790761004906671429919, 2.53655569775089899785250632122, 4.16030512393228373237840730476, 5.20304712749982916182773508495, 6.30866954542946591471542613957, 7.24036024944951791779626823826, 7.80849775331908746972892981901, 9.145382413939012334237956031089, 9.665237736431242843906902789257, 11.05699028783161484141450211078

Graph of the ZZ-function along the critical line