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Properties

Label 2-8325-1.1-c1-0-147
Degree 22
Conductor 83258325
Sign −1-1
Analytic cond. 66.475466.4754
Root an. cond. 8.153248.15324
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  − 1.61·2-s + 0.618·4-s − 0.671·7-s + 2.23·8-s + 3.14·11-s − 5.09·13-s + 1.08·14-s − 4.85·16-s − 5.91·17-s − 0.179·19-s − 5.09·22-s + 4.89·23-s + 8.24·26-s − 0.415·28-s + 0.430·29-s − 1.70·31-s + 3.38·32-s + 9.57·34-s − 37-s + 0.289·38-s + 11.0·41-s + 3.00·43-s + 1.94·44-s − 7.91·46-s + 6.48·47-s − 6.54·49-s − 3.14·52-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s − 0.253·7-s + 0.790·8-s + 0.949·11-s − 1.41·13-s + 0.290·14-s − 1.21·16-s − 1.43·17-s − 0.0411·19-s − 1.08·22-s + 1.02·23-s + 1.61·26-s − 0.0784·28-s + 0.0800·29-s − 0.306·31-s + 0.597·32-s + 1.64·34-s − 0.164·37-s + 0.0470·38-s + 1.73·41-s + 0.458·43-s + 0.293·44-s − 1.16·46-s + 0.945·47-s − 0.935·49-s − 0.436·52-s + ⋯

Functional equation

Λ(s)=(8325s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8325s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 83258325    =    32⋅52⋅373^{2} \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 66.475466.4754
Root analytic conductor: 8.153248.15324
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8325, ( :1/2), −1)(2,\ 8325,\ (\ :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
37 1+T 1 + T
good2 1+1.61T+2T2 1 + 1.61T + 2T^{2}
7 1+0.671T+7T2 1 + 0.671T + 7T^{2}
11 1−3.14T+11T2 1 - 3.14T + 11T^{2}
13 1+5.09T+13T2 1 + 5.09T + 13T^{2}
17 1+5.91T+17T2 1 + 5.91T + 17T^{2}
19 1+0.179T+19T2 1 + 0.179T + 19T^{2}
23 1−4.89T+23T2 1 - 4.89T + 23T^{2}
29 1−0.430T+29T2 1 - 0.430T + 29T^{2}
31 1+1.70T+31T2 1 + 1.70T + 31T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1−3.00T+43T2 1 - 3.00T + 43T^{2}
47 1−6.48T+47T2 1 - 6.48T + 47T^{2}
53 1−1.64T+53T2 1 - 1.64T + 53T^{2}
59 1−3.40T+59T2 1 - 3.40T + 59T^{2}
61 1−4.00T+61T2 1 - 4.00T + 61T^{2}
67 1+8.44T+67T2 1 + 8.44T + 67T^{2}
71 1−4.41T+71T2 1 - 4.41T + 71T^{2}
73 1+5.40T+73T2 1 + 5.40T + 73T^{2}
79 1−9.01T+79T2 1 - 9.01T + 79T^{2}
83 1+5.82T+83T2 1 + 5.82T + 83T^{2}
89 1+2.45T+89T2 1 + 2.45T + 89T^{2}
97 1+5.56T+97T2 1 + 5.56T + 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

−7.39368504863425106874441063504, −7.05405203660939284282549508926, −6.38830329081664251843882854642, −5.35559983459171870944437886993, −4.52790088424981524522941405532, −4.05930437803014184409804806386, −2.78179482729095516141510465548, −2.06099545097201836895494878665, −1.01124765138665277781201792426, 0, 1.01124765138665277781201792426, 2.06099545097201836895494878665, 2.78179482729095516141510465548, 4.05930437803014184409804806386, 4.52790088424981524522941405532, 5.35559983459171870944437886993, 6.38830329081664251843882854642, 7.05405203660939284282549508926, 7.39368504863425106874441063504

Graph of the ZZ-function along the critical line