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Properties

Label 2-15e2-15.2-c1-0-2
Degree 22
Conductor 225225
Sign 0.998−0.0618i0.998 - 0.0618i
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  + (0.707 − 0.707i)2-s + 0.999i·4-s + (2 + 2i)7-s + (2.12 + 2.12i)8-s − 2.82i·11-s + (−1 + i)13-s + 2.82·14-s + 1.00·16-s + (2.82 − 2.82i)17-s + (−2.00 − 2.00i)22-s + (−2.82 − 2.82i)23-s + 1.41i·26-s + (−1.99 + 1.99i)28-s − 4.24·29-s − 4·31-s + (−3.53 + 3.53i)32-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + 0.499i·4-s + (0.755 + 0.755i)7-s + (0.750 + 0.750i)8-s − 0.852i·11-s + (−0.277 + 0.277i)13-s + 0.755·14-s + 0.250·16-s + (0.685 − 0.685i)17-s + (−0.426 − 0.426i)22-s + (−0.589 − 0.589i)23-s + 0.277i·26-s + (−0.377 + 0.377i)28-s − 0.787·29-s − 0.718·31-s + (−0.624 + 0.624i)32-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.998−0.0618i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.998−0.0618i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.998−0.0618i0.998 - 0.0618i
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), 0.998−0.0618i)(2,\ 225,\ (\ :1/2),\ 0.998 - 0.0618i)

Particular Values

L(1)L(1) ≈\approx 1.66462+0.0515409i1.66462 + 0.0515409i
L(12)L(\frac12) ≈\approx 1.66462+0.0515409i1.66462 + 0.0515409i
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+(−0.707+0.707i)T−2iT2 1 + (-0.707 + 0.707i)T - 2iT^{2}
7 1+(−2−2i)T+7iT2 1 + (-2 - 2i)T + 7iT^{2}
11 1+2.82iT−11T2 1 + 2.82iT - 11T^{2}
13 1+(1−i)T−13iT2 1 + (1 - i)T - 13iT^{2}
17 1+(−2.82+2.82i)T−17iT2 1 + (-2.82 + 2.82i)T - 17iT^{2}
19 1−19T2 1 - 19T^{2}
23 1+(2.82+2.82i)T+23iT2 1 + (2.82 + 2.82i)T + 23iT^{2}
29 1+4.24T+29T2 1 + 4.24T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+(1+i)T+37iT2 1 + (1 + i)T + 37iT^{2}
41 1−1.41iT−41T2 1 - 1.41iT - 41T^{2}
43 1+(−8+8i)T−43iT2 1 + (-8 + 8i)T - 43iT^{2}
47 1+(5.65−5.65i)T−47iT2 1 + (5.65 - 5.65i)T - 47iT^{2}
53 1+(2.82+2.82i)T+53iT2 1 + (2.82 + 2.82i)T + 53iT^{2}
59 1+8.48T+59T2 1 + 8.48T + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+(4+4i)T+67iT2 1 + (4 + 4i)T + 67iT^{2}
71 1−5.65iT−71T2 1 - 5.65iT - 71T^{2}
73 1+(1−i)T−73iT2 1 + (1 - i)T - 73iT^{2}
79 1+12iT−79T2 1 + 12iT - 79T^{2}
83 1+(2.82+2.82i)T+83iT2 1 + (2.82 + 2.82i)T + 83iT^{2}
89 1−12.7T+89T2 1 - 12.7T + 89T^{2}
97 1+(−11−11i)T+97iT2 1 + (-11 - 11i)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

−12.12361151186186338459818859337, −11.53934067007424217991385147819, −10.69460255020947430415038567067, −9.226469386394698919591401633144, −8.299540793805372021990532734293, −7.41439774919426906661234688227, −5.77736108683814380360010462353, −4.76327402091490315405806228426, −3.43575388767802316861762507108, −2.15141207381034754334356551336, 1.60237081858699516247308595620, 3.92599680435465843769319585591, 4.92528642053359984071928104512, 5.94259361664127967526430284693, 7.22408914352540503292374627314, 7.87402606288572129739644673693, 9.509366422982229982987214642022, 10.32011556253008692366435768901, 11.16369469681601332877654522516, 12.44350411956089834311056661805

Graph of the ZZ-function along the critical line