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Properties

Label 2-425-1.1-c1-0-19
Degree 22
Conductor 425425
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 0.585·3-s + 3.82·4-s + 1.41·6-s + 3.41·7-s + 4.41·8-s − 2.65·9-s − 5.41·11-s + 2.24·12-s − 2.82·13-s + 8.24·14-s + 2.99·16-s + 17-s − 6.41·18-s + 2.82·19-s + 2·21-s − 13.0·22-s + 0.585·23-s + 2.58·24-s − 6.82·26-s − 3.31·27-s + 13.0·28-s + 0.828·29-s − 4.24·31-s − 1.58·32-s − 3.17·33-s + 2.41·34-s + ⋯
L(s)  = 1  + 1.70·2-s + 0.338·3-s + 1.91·4-s + 0.577·6-s + 1.29·7-s + 1.56·8-s − 0.885·9-s − 1.63·11-s + 0.647·12-s − 0.784·13-s + 2.20·14-s + 0.749·16-s + 0.242·17-s − 1.51·18-s + 0.648·19-s + 0.436·21-s − 2.78·22-s + 0.122·23-s + 0.527·24-s − 1.33·26-s − 0.637·27-s + 2.47·28-s + 0.153·29-s − 0.762·31-s − 0.280·32-s − 0.552·33-s + 0.414·34-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: 11
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: 00
Selberg data: (2, 425, ( :1/2), 1)(2,\ 425,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.7425238633.742523863
L(12)L(\frac12) ≈\approx 3.7425238633.742523863
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 1−2.41T+2T2 1 - 2.41T + 2T^{2}
3 1−0.585T+3T2 1 - 0.585T + 3T^{2}
7 1−3.41T+7T2 1 - 3.41T + 7T^{2}
11 1+5.41T+11T2 1 + 5.41T + 11T^{2}
13 1+2.82T+13T2 1 + 2.82T + 13T^{2}
19 1−2.82T+19T2 1 - 2.82T + 19T^{2}
23 1−0.585T+23T2 1 - 0.585T + 23T^{2}
29 1−0.828T+29T2 1 - 0.828T + 29T^{2}
31 1+4.24T+31T2 1 + 4.24T + 31T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1−10.4T+41T2 1 - 10.4T + 41T^{2}
43 1−3.65T+43T2 1 - 3.65T + 43T^{2}
47 1+0.828T+47T2 1 + 0.828T + 47T^{2}
53 1+11.6T+53T2 1 + 11.6T + 53T^{2}
59 1+14.8T+59T2 1 + 14.8T + 59T^{2}
61 1+3.65T+61T2 1 + 3.65T + 61T^{2}
67 1−8.82T+67T2 1 - 8.82T + 67T^{2}
71 1−4.24T+71T2 1 - 4.24T + 71T^{2}
73 1+0.828T+73T2 1 + 0.828T + 73T^{2}
79 1−2.58T+79T2 1 - 2.58T + 79T^{2}
83 1−13.3T+83T2 1 - 13.3T + 83T^{2}
89 1+13.6T+89T2 1 + 13.6T + 89T^{2}
97 1−7.65T+97T2 1 - 7.65T + 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

−11.23664520772967716513190019753, −10.90577701985937228495745063114, −9.411322287712602102775235249481, −7.931599288782309644467158872189, −7.58492659120522843879784448101, −5.95927652893199961786478029183, −5.21187388188051352096779670999, −4.55161728048305662732947370380, −3.06745154060573483765755880299, −2.28272312873032065049441638696, 2.28272312873032065049441638696, 3.06745154060573483765755880299, 4.55161728048305662732947370380, 5.21187388188051352096779670999, 5.95927652893199961786478029183, 7.58492659120522843879784448101, 7.931599288782309644467158872189, 9.411322287712602102775235249481, 10.90577701985937228495745063114, 11.23664520772967716513190019753

Graph of the ZZ-function along the critical line