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Properties

Label 2-297-297.175-c0-0-0
Degree 22
Conductor 297297
Sign −0.448+0.893i-0.448 + 0.893i
Analytic cond. 0.1482220.148222
Root an. cond. 0.3849960.384996
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)5-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (−1.43 + 1.20i)15-s + (0.766 − 0.642i)16-s + (1.76 + 0.642i)20-s + (0.939 − 0.342i)23-s + (0.439 + 2.49i)25-s + (−0.5 + 0.866i)27-s + (−0.326 + 0.118i)31-s + (−0.5 − 0.866i)33-s + 0.999·36-s + (0.939 − 1.62i)37-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)5-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (−1.43 + 1.20i)15-s + (0.766 − 0.642i)16-s + (1.76 + 0.642i)20-s + (0.939 − 0.342i)23-s + (0.439 + 2.49i)25-s + (−0.5 + 0.866i)27-s + (−0.326 + 0.118i)31-s + (−0.5 − 0.866i)33-s + 0.999·36-s + (0.939 − 1.62i)37-s + ⋯

Functional equation

Λ(s)=(297s/2 ΓC(s) L(s)=((−0.448+0.893i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(297s/2 ΓC(s) L(s)=((−0.448+0.893i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 297297    =    33⋅113^{3} \cdot 11
Sign: −0.448+0.893i-0.448 + 0.893i
Analytic conductor: 0.1482220.148222
Root analytic conductor: 0.3849960.384996
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ297(175,⋅)\chi_{297} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 297, ( :0), −0.448+0.893i)(2,\ 297,\ (\ :0),\ -0.448 + 0.893i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50638216240.5063821624
L(12)L(\frac12) ≈\approx 0.50638216240.5063821624
L(1)L(1) 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)
bad3 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
11 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
good2 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
5 1+(1.43+1.20i)T+(0.173+0.984i)T2 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2}
7 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
13 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
29 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
31 1+(0.326−0.118i)T+(0.766−0.642i)T2 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2}
37 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
47 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
53 1−0.347T+T2 1 - 0.347T + T^{2}
59 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
61 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
67 1+(−0.0603+0.342i)T+(−0.939−0.342i)T2 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2}
71 1+(0.766−1.32i)T+(−0.5−0.866i)T2 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
83 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
89 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+(−1.17+0.984i)T+(0.173−0.984i)T2 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)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

−11.91396847228151103016869557294, −11.19695650726369249432850093526, −9.220039328163842485012296009008, −8.715545822993928387111914515755, −8.008646794304444163677104053839, −7.14468547815721637488562187364, −5.55918773180648514914797225710, −4.36598838527626367752588302057, −3.40459537878854720293399672875, −0.864307895157398535155276217590, 3.16794659852556745005572638295, 4.04047450003798566291325691540, 4.86139087454397805016920937725, 6.43120060138251007139008625511, 7.63821013592983897437007039599, 8.588258130273881292479443876364, 9.579274651870016506474292225442, 10.38469081680986122858317922659, 11.25171570023346297148621185333, 11.96584597253416399555273709629

Graph of the ZZ-function along the critical line