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Properties

Label 2-425-1.1-c1-0-13
Degree 22
Conductor 425425
Sign −1-1
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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  − 2.31·2-s − 0.203·3-s + 3.36·4-s + 0.470·6-s − 0.683·7-s − 3.16·8-s − 2.95·9-s + 3.68·11-s − 0.683·12-s − 4.43·13-s + 1.58·14-s + 0.593·16-s + 17-s + 6.85·18-s − 1.03·19-s + 0.138·21-s − 8.52·22-s + 4.52·23-s + 0.642·24-s + 10.2·26-s + 1.21·27-s − 2.30·28-s − 3.69·29-s − 10.8·31-s + 4.94·32-s − 0.747·33-s − 2.31·34-s + ⋯
L(s)  = 1  − 1.63·2-s − 0.117·3-s + 1.68·4-s + 0.192·6-s − 0.258·7-s − 1.11·8-s − 0.986·9-s + 1.10·11-s − 0.197·12-s − 1.23·13-s + 0.423·14-s + 0.148·16-s + 0.242·17-s + 1.61·18-s − 0.238·19-s + 0.0303·21-s − 1.81·22-s + 0.943·23-s + 0.131·24-s + 2.01·26-s + 0.233·27-s − 0.434·28-s − 0.685·29-s − 1.95·31-s + 0.874·32-s − 0.130·33-s − 0.397·34-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −1-1
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 425, ( :1/2), −1)(2,\ 425,\ (\ :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)
bad5 1 1
17 1−T 1 - T
good2 1+2.31T+2T2 1 + 2.31T + 2T^{2}
3 1+0.203T+3T2 1 + 0.203T + 3T^{2}
7 1+0.683T+7T2 1 + 0.683T + 7T^{2}
11 1−3.68T+11T2 1 - 3.68T + 11T^{2}
13 1+4.43T+13T2 1 + 4.43T + 13T^{2}
19 1+1.03T+19T2 1 + 1.03T + 19T^{2}
23 1−4.52T+23T2 1 - 4.52T + 23T^{2}
29 1+3.69T+29T2 1 + 3.69T + 29T^{2}
31 1+10.8T+31T2 1 + 10.8T + 31T^{2}
37 1+0.308T+37T2 1 + 0.308T + 37T^{2}
41 1+6.15T+41T2 1 + 6.15T + 41T^{2}
43 1+7.88T+43T2 1 + 7.88T + 43T^{2}
47 1+4.43T+47T2 1 + 4.43T + 47T^{2}
53 1+11.4T+53T2 1 + 11.4T + 53T^{2}
59 1+2T+59T2 1 + 2T + 59T^{2}
61 1−9.94T+61T2 1 - 9.94T + 61T^{2}
67 1+9.16T+67T2 1 + 9.16T + 67T^{2}
71 1+9.37T+71T2 1 + 9.37T + 71T^{2}
73 1−2.26T+73T2 1 - 2.26T + 73T^{2}
79 1−7.42T+79T2 1 - 7.42T + 79T^{2}
83 1−8.92T+83T2 1 - 8.92T + 83T^{2}
89 1−11.5T+89T2 1 - 11.5T + 89T^{2}
97 1−12.5T+97T2 1 - 12.5T + 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

−10.58704147183724757622914305460, −9.506704808080261372102959757449, −9.136492936850092882338971785650, −8.151994830205477033804214503623, −7.19571768261755375721633500241, −6.41937338242797308676389304995, −5.07519971371478406819503938213, −3.27142538609352599865393427468, −1.80943100643295076483839057996, 0, 1.80943100643295076483839057996, 3.27142538609352599865393427468, 5.07519971371478406819503938213, 6.41937338242797308676389304995, 7.19571768261755375721633500241, 8.151994830205477033804214503623, 9.136492936850092882338971785650, 9.506704808080261372102959757449, 10.58704147183724757622914305460

Graph of the ZZ-function along the critical line