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Properties

Label 2-7623-1.1-c1-0-209
Degree 22
Conductor 76237623
Sign −1-1
Analytic cond. 60.869960.8699
Root an. cond. 7.801927.80192
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  + 0.618·2-s − 1.61·4-s + 0.381·5-s + 7-s − 2.23·8-s + 0.236·10-s − 13-s + 0.618·14-s + 1.85·16-s + 4.23·17-s − 0.618·20-s + 3.23·23-s − 4.85·25-s − 0.618·26-s − 1.61·28-s − 6.70·29-s − 10.2·31-s + 5.61·32-s + 2.61·34-s + 0.381·35-s + 6.94·37-s − 0.854·40-s − 5.09·41-s − 43-s + 2.00·46-s + 7.32·47-s + 49-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s + 0.170·5-s + 0.377·7-s − 0.790·8-s + 0.0746·10-s − 0.277·13-s + 0.165·14-s + 0.463·16-s + 1.02·17-s − 0.138·20-s + 0.674·23-s − 0.970·25-s − 0.121·26-s − 0.305·28-s − 1.24·29-s − 1.83·31-s + 0.993·32-s + 0.448·34-s + 0.0645·35-s + 1.14·37-s − 0.135·40-s − 0.794·41-s − 0.152·43-s + 0.294·46-s + 1.06·47-s + 0.142·49-s + ⋯

Functional equation

Λ(s)=(7623s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7623s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76237623    =    32⋅7⋅1123^{2} \cdot 7 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 60.869960.8699
Root analytic conductor: 7.801927.80192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7623, ( :1/2), −1)(2,\ 7623,\ (\ :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
7 1−T 1 - T
11 1 1
good2 1−0.618T+2T2 1 - 0.618T + 2T^{2}
5 1−0.381T+5T2 1 - 0.381T + 5T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1−4.23T+17T2 1 - 4.23T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−3.23T+23T2 1 - 3.23T + 23T^{2}
29 1+6.70T+29T2 1 + 6.70T + 29T^{2}
31 1+10.2T+31T2 1 + 10.2T + 31T^{2}
37 1−6.94T+37T2 1 - 6.94T + 37T^{2}
41 1+5.09T+41T2 1 + 5.09T + 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1−7.32T+47T2 1 - 7.32T + 47T^{2}
53 1+7.61T+53T2 1 + 7.61T + 53T^{2}
59 1−4.14T+59T2 1 - 4.14T + 59T^{2}
61 1+5.76T+61T2 1 + 5.76T + 61T^{2}
67 1+9.23T+67T2 1 + 9.23T + 67T^{2}
71 1−7.47T+71T2 1 - 7.47T + 71T^{2}
73 1−11.5T+73T2 1 - 11.5T + 73T^{2}
79 1−10.8T+79T2 1 - 10.8T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−6.38T+89T2 1 - 6.38T + 89T^{2}
97 1+17T+97T2 1 + 17T + 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

−7.74607574496326732163732655815, −6.80293717043010550500974724616, −5.79973212324884226975447910529, −5.44421402194162496616611601776, −4.77250187209593489514959980178, −3.87219579206636883680054317876, −3.40356513863813462369537628487, −2.30442816032702091542250214980, −1.26301442173657144658398263786, 0, 1.26301442173657144658398263786, 2.30442816032702091542250214980, 3.40356513863813462369537628487, 3.87219579206636883680054317876, 4.77250187209593489514959980178, 5.44421402194162496616611601776, 5.79973212324884226975447910529, 6.80293717043010550500974724616, 7.74607574496326732163732655815

Graph of the ZZ-function along the critical line