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Properties

Label 2-15e2-15.2-c1-0-4
Degree 22
Conductor 225225
Sign −0.161+0.986i-0.161 + 0.986i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (1.73 − 1.73i)2-s − 3.99i·4-s + (1.22 + 1.22i)7-s + (−3.46 − 3.46i)8-s − 4.24i·11-s + (−3.67 + 3.67i)13-s + 4.24·14-s − 3.99·16-s + (−1.73 + 1.73i)17-s + 5i·19-s + (−7.34 − 7.34i)22-s + (1.73 + 1.73i)23-s + 12.7i·26-s + (4.89 − 4.89i)28-s + 4.24·29-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)2-s − 1.99i·4-s + (0.462 + 0.462i)7-s + (−1.22 − 1.22i)8-s − 1.27i·11-s + (−1.01 + 1.01i)13-s + 1.13·14-s − 0.999·16-s + (−0.420 + 0.420i)17-s + 1.14i·19-s + (−1.56 − 1.56i)22-s + (0.361 + 0.361i)23-s + 2.49i·26-s + (0.925 − 0.925i)28-s + 0.787·29-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.161+0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.161+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.161+0.986i-0.161 + 0.986i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(107,⋅)\chi_{225} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), −0.161+0.986i)(2,\ 225,\ (\ :1/2),\ -0.161 + 0.986i)

Particular Values

L(1)L(1) ≈\approx 1.42254−1.67489i1.42254 - 1.67489i
L(12)L(\frac12) ≈\approx 1.42254−1.67489i1.42254 - 1.67489i
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)
bad3 1 1
5 1 1
good2 1+(−1.73+1.73i)T−2iT2 1 + (-1.73 + 1.73i)T - 2iT^{2}
7 1+(−1.22−1.22i)T+7iT2 1 + (-1.22 - 1.22i)T + 7iT^{2}
11 1+4.24iT−11T2 1 + 4.24iT - 11T^{2}
13 1+(3.67−3.67i)T−13iT2 1 + (3.67 - 3.67i)T - 13iT^{2}
17 1+(1.73−1.73i)T−17iT2 1 + (1.73 - 1.73i)T - 17iT^{2}
19 1−5iT−19T2 1 - 5iT - 19T^{2}
23 1+(−1.73−1.73i)T+23iT2 1 + (-1.73 - 1.73i)T + 23iT^{2}
29 1−4.24T+29T2 1 - 4.24T + 29T^{2}
31 1−T+31T2 1 - T + 31T^{2}
37 1+(−2.44−2.44i)T+37iT2 1 + (-2.44 - 2.44i)T + 37iT^{2}
41 1+8.48iT−41T2 1 + 8.48iT - 41T^{2}
43 1+(1.22−1.22i)T−43iT2 1 + (1.22 - 1.22i)T - 43iT^{2}
47 1+(5.19−5.19i)T−47iT2 1 + (5.19 - 5.19i)T - 47iT^{2}
53 1+(6.92+6.92i)T+53iT2 1 + (6.92 + 6.92i)T + 53iT^{2}
59 1+12.7T+59T2 1 + 12.7T + 59T^{2}
61 1+7T+61T2 1 + 7T + 61T^{2}
67 1+(−3.67−3.67i)T+67iT2 1 + (-3.67 - 3.67i)T + 67iT^{2}
71 1−8.48iT−71T2 1 - 8.48iT - 71T^{2}
73 1+(−2.44+2.44i)T−73iT2 1 + (-2.44 + 2.44i)T - 73iT^{2}
79 1+2iT−79T2 1 + 2iT - 79T^{2}
83 1+(−1.73−1.73i)T+83iT2 1 + (-1.73 - 1.73i)T + 83iT^{2}
89 1−8.48T+89T2 1 - 8.48T + 89T^{2}
97 1+(8.57+8.57i)T+97iT2 1 + (8.57 + 8.57i)T + 97iT^{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

−11.94759769507453077428932928475, −11.35006249867237928513604936421, −10.45506297873959919173887657904, −9.368068875813559934955589780812, −8.145481342643812868966019063657, −6.40061017556961704003843862937, −5.36868942085162070229423723846, −4.34702313679672812887203264441, −3.11484971180110186144680182269, −1.79318744664388340199161078881, 2.85391935217092437432548227444, 4.63472054137443077423453915682, 4.88639721950486722783021450170, 6.43632609983115554070979723455, 7.31977083729115718513766763869, 7.963455720396301469773036376915, 9.459379223035889961518583690411, 10.70833690122989171497862312399, 12.03836381489548060961156198571, 12.76326097835072108155974907559

Graph of the ZZ-function along the critical line