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Properties

Label 2-370-1.1-c1-0-10
Degree 22
Conductor 370370
Sign −1-1
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  − 2-s + 0.732·3-s + 4-s + 5-s − 0.732·6-s − 4.73·7-s − 8-s − 2.46·9-s − 10-s − 5.46·11-s + 0.732·12-s − 5.46·13-s + 4.73·14-s + 0.732·15-s + 16-s + 5.46·17-s + 2.46·18-s + 6.19·19-s + 20-s − 3.46·21-s + 5.46·22-s − 8·23-s − 0.732·24-s + 25-s + 5.46·26-s − 4·27-s − 4.73·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.422·3-s + 0.5·4-s + 0.447·5-s − 0.298·6-s − 1.78·7-s − 0.353·8-s − 0.821·9-s − 0.316·10-s − 1.64·11-s + 0.211·12-s − 1.51·13-s + 1.26·14-s + 0.189·15-s + 0.250·16-s + 1.32·17-s + 0.580·18-s + 1.42·19-s + 0.223·20-s − 0.755·21-s + 1.16·22-s − 1.66·23-s − 0.149·24-s + 0.200·25-s + 1.07·26-s − 0.769·27-s − 0.894·28-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −1-1
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 370, ( :1/2), −1)(2,\ 370,\ (\ :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)
bad2 1+T 1 + T
5 1−T 1 - T
37 1−T 1 - T
good3 1−0.732T+3T2 1 - 0.732T + 3T^{2}
7 1+4.73T+7T2 1 + 4.73T + 7T^{2}
11 1+5.46T+11T2 1 + 5.46T + 11T^{2}
13 1+5.46T+13T2 1 + 5.46T + 13T^{2}
17 1−5.46T+17T2 1 - 5.46T + 17T^{2}
19 1−6.19T+19T2 1 - 6.19T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−4.92T+29T2 1 - 4.92T + 29T^{2}
31 1−0.732T+31T2 1 - 0.732T + 31T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−6.92T+43T2 1 - 6.92T + 43T^{2}
47 1+4.73T+47T2 1 + 4.73T + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1+4.92T+61T2 1 + 4.92T + 61T^{2}
67 1+3.66T+67T2 1 + 3.66T + 67T^{2}
71 1−2.92T+71T2 1 - 2.92T + 71T^{2}
73 1+0.928T+73T2 1 + 0.928T + 73T^{2}
79 1−8.73T+79T2 1 - 8.73T + 79T^{2}
83 1+8.73T+83T2 1 + 8.73T + 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−10.46326174053223918931304521411, −9.806964005969077029406927965912, −9.470504733169353546690605268090, −8.038043063656137002117730924866, −7.45397717871850861724929250526, −6.13419658720380308842395948086, −5.33494467746224220922225621034, −3.13746761846583704719760510993, −2.61882371334183910411617060775, 0, 2.61882371334183910411617060775, 3.13746761846583704719760510993, 5.33494467746224220922225621034, 6.13419658720380308842395948086, 7.45397717871850861724929250526, 8.038043063656137002117730924866, 9.470504733169353546690605268090, 9.806964005969077029406927965912, 10.46326174053223918931304521411

Graph of the ZZ-function along the critical line