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Properties

Label 2-1575-1.1-c1-0-30
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 12.576412.5764
Root an. cond. 3.546323.54632
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 1.61·4-s − 7-s + 2.23·8-s + 0.236·11-s − 1.23·13-s + 0.618·14-s + 1.85·16-s + 2.47·17-s − 4.47·19-s − 0.145·22-s + 6.23·23-s + 0.763·26-s + 1.61·28-s − 5·29-s + 3.70·31-s − 5.61·32-s − 1.52·34-s − 3·37-s + 2.76·38-s − 4.76·41-s − 1.76·43-s − 0.381·44-s − 3.85·46-s − 2·47-s + 49-s + 2.00·52-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 0.377·7-s + 0.790·8-s + 0.0711·11-s − 0.342·13-s + 0.165·14-s + 0.463·16-s + 0.599·17-s − 1.02·19-s − 0.0311·22-s + 1.30·23-s + 0.149·26-s + 0.305·28-s − 0.928·29-s + 0.666·31-s − 0.993·32-s − 0.262·34-s − 0.493·37-s + 0.448·38-s − 0.744·41-s − 0.268·43-s − 0.0575·44-s − 0.568·46-s − 0.291·47-s + 0.142·49-s + 0.277·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 12.576412.5764
Root analytic conductor: 3.546323.54632
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :1/2), −1)(2,\ 1575,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7 1+T 1 + T
good2 1+0.618T+2T2 1 + 0.618T + 2T^{2}
11 1−0.236T+11T2 1 - 0.236T + 11T^{2}
13 1+1.23T+13T2 1 + 1.23T + 13T^{2}
17 1−2.47T+17T2 1 - 2.47T + 17T^{2}
19 1+4.47T+19T2 1 + 4.47T + 19T^{2}
23 1−6.23T+23T2 1 - 6.23T + 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1−3.70T+31T2 1 - 3.70T + 31T^{2}
37 1+3T+37T2 1 + 3T + 37T^{2}
41 1+4.76T+41T2 1 + 4.76T + 41T^{2}
43 1+1.76T+43T2 1 + 1.76T + 43T^{2}
47 1+2T+47T2 1 + 2T + 47T^{2}
53 1−8.47T+53T2 1 - 8.47T + 53T^{2}
59 1+11.7T+59T2 1 + 11.7T + 59T^{2}
61 1+9.70T+61T2 1 + 9.70T + 61T^{2}
67 1−4.23T+67T2 1 - 4.23T + 67T^{2}
71 1+8.70T+71T2 1 + 8.70T + 71T^{2}
73 1−8.76T+73T2 1 - 8.76T + 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1+7.70T+83T2 1 + 7.70T + 83T^{2}
89 1+17.2T+89T2 1 + 17.2T + 89T^{2}
97 1+5.23T+97T2 1 + 5.23T + 97T^{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

−9.015818767480641590494540269519, −8.414233578723368141617527311173, −7.53148923264926700975358271651, −6.75009883347375055028335954907, −5.66712829094775006233928476008, −4.84122488701842089366980507762, −3.96703907466446787406158731692, −2.93410291197492932117551952451, −1.44631677378036851450710791691, 0, 1.44631677378036851450710791691, 2.93410291197492932117551952451, 3.96703907466446787406158731692, 4.84122488701842089366980507762, 5.66712829094775006233928476008, 6.75009883347375055028335954907, 7.53148923264926700975358271651, 8.414233578723368141617527311173, 9.015818767480641590494540269519

Graph of the ZZ-function along the critical line