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Properties

Label 2-95e2-1.1-c1-0-398
Degree 22
Conductor 90259025
Sign −1-1
Analytic cond. 72.064972.0649
Root an. cond. 8.489108.48910
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  − 2.17·2-s + 1.41·3-s + 2.70·4-s − 3.06·6-s + 1.76·7-s − 1.53·8-s − 1.00·9-s + 1.83·11-s + 3.82·12-s + 2.60·13-s − 3.82·14-s − 2.07·16-s − 4.23·17-s + 2.17·18-s + 2.48·21-s − 3.98·22-s + 2.20·23-s − 2.17·24-s − 5.65·26-s − 5.65·27-s + 4.77·28-s − 7.12·29-s + 0.303·31-s + 7.58·32-s + 2.59·33-s + 9.19·34-s − 2.72·36-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.815·3-s + 1.35·4-s − 1.25·6-s + 0.665·7-s − 0.544·8-s − 0.334·9-s + 0.554·11-s + 1.10·12-s + 0.722·13-s − 1.02·14-s − 0.519·16-s − 1.02·17-s + 0.513·18-s + 0.543·21-s − 0.850·22-s + 0.459·23-s − 0.443·24-s − 1.10·26-s − 1.08·27-s + 0.902·28-s − 1.32·29-s + 0.0545·31-s + 1.34·32-s + 0.452·33-s + 1.57·34-s − 0.453·36-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 72.064972.0649
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9025, ( :1/2), −1)(2,\ 9025,\ (\ :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
19 1 1
good2 1+2.17T+2T2 1 + 2.17T + 2T^{2}
3 1−1.41T+3T2 1 - 1.41T + 3T^{2}
7 1−1.76T+7T2 1 - 1.76T + 7T^{2}
11 1−1.83T+11T2 1 - 1.83T + 11T^{2}
13 1−2.60T+13T2 1 - 2.60T + 13T^{2}
17 1+4.23T+17T2 1 + 4.23T + 17T^{2}
23 1−2.20T+23T2 1 - 2.20T + 23T^{2}
29 1+7.12T+29T2 1 + 7.12T + 29T^{2}
31 1−0.303T+31T2 1 - 0.303T + 31T^{2}
37 1+3.90T+37T2 1 + 3.90T + 37T^{2}
41 1−8.23T+41T2 1 - 8.23T + 41T^{2}
43 1+2.34T+43T2 1 + 2.34T + 43T^{2}
47 1−7.25T+47T2 1 - 7.25T + 47T^{2}
53 1+10.6T+53T2 1 + 10.6T + 53T^{2}
59 1+12.0T+59T2 1 + 12.0T + 59T^{2}
61 1−10.5T+61T2 1 - 10.5T + 61T^{2}
67 1+13.0T+67T2 1 + 13.0T + 67T^{2}
71 1−11.8T+71T2 1 - 11.8T + 71T^{2}
73 1−9.16T+73T2 1 - 9.16T + 73T^{2}
79 1−7.88T+79T2 1 - 7.88T + 79T^{2}
83 1+6.93T+83T2 1 + 6.93T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+7.75T+97T2 1 + 7.75T + 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

−7.76146004534449371008594931157, −6.99577866705735556029100078398, −6.35048419859340212909899857638, −5.45903684032668128762472405691, −4.44698210388984906393047338145, −3.70264609540462482961027922819, −2.71381159475707967836875432660, −1.93853977782619403454249606637, −1.27766268809669061371292809971, 0, 1.27766268809669061371292809971, 1.93853977782619403454249606637, 2.71381159475707967836875432660, 3.70264609540462482961027922819, 4.44698210388984906393047338145, 5.45903684032668128762472405691, 6.35048419859340212909899857638, 6.99577866705735556029100078398, 7.76146004534449371008594931157

Graph of the ZZ-function along the critical line