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Properties

Label 2-350-175.108-c1-0-11
Degree 22
Conductor 350350
Sign 0.726+0.687i0.726 + 0.687i
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  + (−0.0523 − 0.998i)2-s + (0.430 + 0.348i)3-s + (−0.994 + 0.104i)4-s + (2.13 − 0.656i)5-s + (0.325 − 0.448i)6-s + (−0.343 + 2.62i)7-s + (0.156 + 0.987i)8-s + (−0.559 − 2.63i)9-s + (−0.767 − 2.10i)10-s + (4.51 + 0.959i)11-s + (−0.464 − 0.301i)12-s + (−1.77 − 3.47i)13-s + (2.63 + 0.205i)14-s + (1.14 + 0.462i)15-s + (0.978 − 0.207i)16-s + (7.52 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.0370 − 0.706i)2-s + (0.248 + 0.201i)3-s + (−0.497 + 0.0522i)4-s + (0.955 − 0.293i)5-s + (0.132 − 0.182i)6-s + (−0.129 + 0.991i)7-s + (0.0553 + 0.349i)8-s + (−0.186 − 0.878i)9-s + (−0.242 − 0.664i)10-s + (1.36 + 0.289i)11-s + (−0.134 − 0.0870i)12-s + (−0.491 − 0.965i)13-s + (0.704 + 0.0548i)14-s + (0.296 + 0.119i)15-s + (0.244 − 0.0519i)16-s + (1.82 + 0.700i)17-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.726+0.687i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((0.726+0.687i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.726+0.687i0.726 + 0.687i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(283,⋅)\chi_{350} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), 0.726+0.687i)(2,\ 350,\ (\ :1/2),\ 0.726 + 0.687i)

Particular Values

L(1)L(1) ≈\approx 1.49301−0.594567i1.49301 - 0.594567i
L(12)L(\frac12) ≈\approx 1.49301−0.594567i1.49301 - 0.594567i
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+(0.0523+0.998i)T 1 + (0.0523 + 0.998i)T
5 1+(−2.13+0.656i)T 1 + (-2.13 + 0.656i)T
7 1+(0.343−2.62i)T 1 + (0.343 - 2.62i)T
good3 1+(−0.430−0.348i)T+(0.623+2.93i)T2 1 + (-0.430 - 0.348i)T + (0.623 + 2.93i)T^{2}
11 1+(−4.51−0.959i)T+(10.0+4.47i)T2 1 + (-4.51 - 0.959i)T + (10.0 + 4.47i)T^{2}
13 1+(1.77+3.47i)T+(−7.64+10.5i)T2 1 + (1.77 + 3.47i)T + (-7.64 + 10.5i)T^{2}
17 1+(−7.52−2.88i)T+(12.6+11.3i)T2 1 + (-7.52 - 2.88i)T + (12.6 + 11.3i)T^{2}
19 1+(0.203−1.93i)T+(−18.5−3.95i)T2 1 + (0.203 - 1.93i)T + (-18.5 - 3.95i)T^{2}
23 1+(5.87−0.308i)T+(22.8−2.40i)T2 1 + (5.87 - 0.308i)T + (22.8 - 2.40i)T^{2}
29 1+(2.48+3.42i)T+(−8.96+27.5i)T2 1 + (2.48 + 3.42i)T + (-8.96 + 27.5i)T^{2}
31 1+(1.81+4.08i)T+(−20.7+23.0i)T2 1 + (1.81 + 4.08i)T + (-20.7 + 23.0i)T^{2}
37 1+(0.822−1.26i)T+(−15.0−33.8i)T2 1 + (0.822 - 1.26i)T + (-15.0 - 33.8i)T^{2}
41 1+(3.28+1.06i)T+(33.1+24.0i)T2 1 + (3.28 + 1.06i)T + (33.1 + 24.0i)T^{2}
43 1+(−6.54−6.54i)T+43iT2 1 + (-6.54 - 6.54i)T + 43iT^{2}
47 1+(0.294+0.766i)T+(−34.9+31.4i)T2 1 + (0.294 + 0.766i)T + (-34.9 + 31.4i)T^{2}
53 1+(7.12−8.79i)T+(−11.0−51.8i)T2 1 + (7.12 - 8.79i)T + (-11.0 - 51.8i)T^{2}
59 1+(2.32+2.57i)T+(−6.16+58.6i)T2 1 + (2.32 + 2.57i)T + (-6.16 + 58.6i)T^{2}
61 1+(1.38+1.24i)T+(6.37+60.6i)T2 1 + (1.38 + 1.24i)T + (6.37 + 60.6i)T^{2}
67 1+(3.59−9.36i)T+(−49.7−44.8i)T2 1 + (3.59 - 9.36i)T + (-49.7 - 44.8i)T^{2}
71 1+(−2.57+1.87i)T+(21.9−67.5i)T2 1 + (-2.57 + 1.87i)T + (21.9 - 67.5i)T^{2}
73 1+(6.54−4.25i)T+(29.6−66.6i)T2 1 + (6.54 - 4.25i)T + (29.6 - 66.6i)T^{2}
79 1+(3.15−7.09i)T+(−52.8−58.7i)T2 1 + (3.15 - 7.09i)T + (-52.8 - 58.7i)T^{2}
83 1+(14.5−2.31i)T+(78.9−25.6i)T2 1 + (14.5 - 2.31i)T + (78.9 - 25.6i)T^{2}
89 1+(3.16−3.51i)T+(−9.30−88.5i)T2 1 + (3.16 - 3.51i)T + (-9.30 - 88.5i)T^{2}
97 1+(5.61+0.888i)T+(92.2+29.9i)T2 1 + (5.61 + 0.888i)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

−11.59761415169156160138487601845, −10.00944407598257260351927797628, −9.756107104082823959105495707813, −8.918644257516669936586396839147, −7.933734962795924259742375959949, −6.11208786244167529587797425718, −5.64583094068120863465720873022, −4.03587042036957720794744171017, −2.90379927730270667713351728610, −1.49330699402693616914697273146, 1.61523826911282662026086092630, 3.42550679629079010179243523821, 4.79341112408202053453901081830, 5.91859916135748504339443442480, 6.95402644179543868360973918169, 7.53513298283991619873806789022, 8.844617734407687306871085189233, 9.665928035416991291547415355027, 10.41982248204371547475236158309, 11.57754010322994693506356095547

Graph of the ZZ-function along the critical line