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Properties

Label 2-350-175.103-c1-0-11
Degree 22
Conductor 350350
Sign 0.921−0.388i0.921 - 0.388i
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.998 + 0.0523i)2-s + (0.555 + 0.686i)3-s + (0.994 + 0.104i)4-s + (2.14 − 0.642i)5-s + (0.519 + 0.714i)6-s + (0.427 + 2.61i)7-s + (0.987 + 0.156i)8-s + (0.461 − 2.17i)9-s + (2.17 − 0.529i)10-s + (−3.71 + 0.789i)11-s + (0.480 + 0.740i)12-s + (−1.44 − 0.737i)13-s + (0.290 + 2.62i)14-s + (1.63 + 1.11i)15-s + (0.978 + 0.207i)16-s + (−2.38 − 6.21i)17-s + ⋯
L(s)  = 1  + (0.706 + 0.0370i)2-s + (0.320 + 0.396i)3-s + (0.497 + 0.0522i)4-s + (0.957 − 0.287i)5-s + (0.211 + 0.291i)6-s + (0.161 + 0.986i)7-s + (0.349 + 0.0553i)8-s + (0.153 − 0.723i)9-s + (0.686 − 0.167i)10-s + (−1.11 + 0.237i)11-s + (0.138 + 0.213i)12-s + (−0.401 − 0.204i)13-s + (0.0776 + 0.702i)14-s + (0.421 + 0.287i)15-s + (0.244 + 0.0519i)16-s + (−0.578 − 1.50i)17-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.921−0.388i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((0.921−0.388i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.921−0.388i0.921 - 0.388i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(103,⋅)\chi_{350} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), 0.921−0.388i)(2,\ 350,\ (\ :1/2),\ 0.921 - 0.388i)

Particular Values

L(1)L(1) ≈\approx 2.37227+0.479328i2.37227 + 0.479328i
L(12)L(\frac12) ≈\approx 2.37227+0.479328i2.37227 + 0.479328i
L(32)L(\frac{3}{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+(−0.998−0.0523i)T 1 + (-0.998 - 0.0523i)T
5 1+(−2.14+0.642i)T 1 + (-2.14 + 0.642i)T
7 1+(−0.427−2.61i)T 1 + (-0.427 - 2.61i)T
good3 1+(−0.555−0.686i)T+(−0.623+2.93i)T2 1 + (-0.555 - 0.686i)T + (-0.623 + 2.93i)T^{2}
11 1+(3.71−0.789i)T+(10.0−4.47i)T2 1 + (3.71 - 0.789i)T + (10.0 - 4.47i)T^{2}
13 1+(1.44+0.737i)T+(7.64+10.5i)T2 1 + (1.44 + 0.737i)T + (7.64 + 10.5i)T^{2}
17 1+(2.38+6.21i)T+(−12.6+11.3i)T2 1 + (2.38 + 6.21i)T + (-12.6 + 11.3i)T^{2}
19 1+(−0.889−8.46i)T+(−18.5+3.95i)T2 1 + (-0.889 - 8.46i)T + (-18.5 + 3.95i)T^{2}
23 1+(−0.128+2.45i)T+(−22.8−2.40i)T2 1 + (-0.128 + 2.45i)T + (-22.8 - 2.40i)T^{2}
29 1+(3.41−4.69i)T+(−8.96−27.5i)T2 1 + (3.41 - 4.69i)T + (-8.96 - 27.5i)T^{2}
31 1+(−2.15+4.84i)T+(−20.7−23.0i)T2 1 + (-2.15 + 4.84i)T + (-20.7 - 23.0i)T^{2}
37 1+(5.50−3.57i)T+(15.0−33.8i)T2 1 + (5.50 - 3.57i)T + (15.0 - 33.8i)T^{2}
41 1+(−4.80+1.56i)T+(33.1−24.0i)T2 1 + (-4.80 + 1.56i)T + (33.1 - 24.0i)T^{2}
43 1+(3.22+3.22i)T+43iT2 1 + (3.22 + 3.22i)T + 43iT^{2}
47 1+(3.04+1.16i)T+(34.9+31.4i)T2 1 + (3.04 + 1.16i)T + (34.9 + 31.4i)T^{2}
53 1+(6.74−5.46i)T+(11.0−51.8i)T2 1 + (6.74 - 5.46i)T + (11.0 - 51.8i)T^{2}
59 1+(0.404−0.448i)T+(−6.16−58.6i)T2 1 + (0.404 - 0.448i)T + (-6.16 - 58.6i)T^{2}
61 1+(−8.88+8.00i)T+(6.37−60.6i)T2 1 + (-8.88 + 8.00i)T + (6.37 - 60.6i)T^{2}
67 1+(1.26−0.486i)T+(49.7−44.8i)T2 1 + (1.26 - 0.486i)T + (49.7 - 44.8i)T^{2}
71 1+(10.6+7.74i)T+(21.9+67.5i)T2 1 + (10.6 + 7.74i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.723+1.11i)T+(−29.6−66.6i)T2 1 + (-0.723 + 1.11i)T + (-29.6 - 66.6i)T^{2}
79 1+(−1.89−4.26i)T+(−52.8+58.7i)T2 1 + (-1.89 - 4.26i)T + (-52.8 + 58.7i)T^{2}
83 1+(−2.58+16.3i)T+(−78.9−25.6i)T2 1 + (-2.58 + 16.3i)T + (-78.9 - 25.6i)T^{2}
89 1+(−6.29−6.98i)T+(−9.30+88.5i)T2 1 + (-6.29 - 6.98i)T + (-9.30 + 88.5i)T^{2}
97 1+(−0.588−3.71i)T+(−92.2+29.9i)T2 1 + (-0.588 - 3.71i)T + (-92.2 + 29.9i)T^{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

−11.86175717274142033411303293706, −10.48531200322558302508010178786, −9.724783509951492564769222750850, −8.922869055985382082121709224648, −7.77646486370257008871566782544, −6.44336485667945651472065919325, −5.47899360245567146460109744783, −4.77604397393338529361178644774, −3.16796375198277516727084060009, −2.12810983331789873564151860407, 1.84868099339633100750364573995, 2.91127964810471982508446928832, 4.50935251033707953430284243503, 5.42698678911136574837848538826, 6.69251242138219954106064540456, 7.40158875609741393253238426759, 8.451628434985618017128155396229, 9.846753397510883342245779891648, 10.70756912201733783483913025227, 11.18580419292840860236946381386

Graph of the ZZ-function along the critical line