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Properties

Label 2-44e2-1.1-c1-0-23
Degree 22
Conductor 19361936
Sign 11
Analytic cond. 15.459015.4590
Root an. cond. 3.931793.93179
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  + 2.04·3-s − 0.163·5-s + 3.50·7-s + 1.18·9-s − 2.69·13-s − 0.335·15-s + 7.57·17-s + 4.61·19-s + 7.16·21-s + 0.706·23-s − 4.97·25-s − 3.70·27-s + 1.02·29-s − 5.39·31-s − 0.573·35-s + 11.5·37-s − 5.51·39-s + 0.280·41-s + 4.31·43-s − 0.194·45-s − 5.82·47-s + 5.25·49-s + 15.5·51-s + 9.87·53-s + 9.45·57-s + 3.09·59-s − 6.93·61-s + ⋯
L(s)  = 1  + 1.18·3-s − 0.0732·5-s + 1.32·7-s + 0.396·9-s − 0.747·13-s − 0.0865·15-s + 1.83·17-s + 1.05·19-s + 1.56·21-s + 0.147·23-s − 0.994·25-s − 0.713·27-s + 0.191·29-s − 0.969·31-s − 0.0969·35-s + 1.90·37-s − 0.882·39-s + 0.0438·41-s + 0.657·43-s − 0.0290·45-s − 0.850·47-s + 0.751·49-s + 2.17·51-s + 1.35·53-s + 1.25·57-s + 0.402·59-s − 0.887·61-s + ⋯

Functional equation

Λ(s)=(1936s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1936s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19361936    =    24⋅1122^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 15.459015.4590
Root analytic conductor: 3.931793.93179
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1936, ( :1/2), 1)(2,\ 1936,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0921797313.092179731
L(12)L(\frac12) ≈\approx 3.0921797313.092179731
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)
bad2 1 1
11 1 1
good3 1−2.04T+3T2 1 - 2.04T + 3T^{2}
5 1+0.163T+5T2 1 + 0.163T + 5T^{2}
7 1−3.50T+7T2 1 - 3.50T + 7T^{2}
13 1+2.69T+13T2 1 + 2.69T + 13T^{2}
17 1−7.57T+17T2 1 - 7.57T + 17T^{2}
19 1−4.61T+19T2 1 - 4.61T + 19T^{2}
23 1−0.706T+23T2 1 - 0.706T + 23T^{2}
29 1−1.02T+29T2 1 - 1.02T + 29T^{2}
31 1+5.39T+31T2 1 + 5.39T + 31T^{2}
37 1−11.5T+37T2 1 - 11.5T + 37T^{2}
41 1−0.280T+41T2 1 - 0.280T + 41T^{2}
43 1−4.31T+43T2 1 - 4.31T + 43T^{2}
47 1+5.82T+47T2 1 + 5.82T + 47T^{2}
53 1−9.87T+53T2 1 - 9.87T + 53T^{2}
59 1−3.09T+59T2 1 - 3.09T + 59T^{2}
61 1+6.93T+61T2 1 + 6.93T + 61T^{2}
67 1+1.91T+67T2 1 + 1.91T + 67T^{2}
71 1+12.1T+71T2 1 + 12.1T + 71T^{2}
73 1−11.3T+73T2 1 - 11.3T + 73T^{2}
79 1+1.72T+79T2 1 + 1.72T + 79T^{2}
83 1−4.70T+83T2 1 - 4.70T + 83T^{2}
89 1−6.31T+89T2 1 - 6.31T + 89T^{2}
97 1−7.00T+97T2 1 - 7.00T + 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

−9.255412632096169883997497117505, −8.167786562522512224681200743672, −7.78487293444204153358534125159, −7.34927148444945221696381934045, −5.79556690961774165079391688764, −5.15826309875877454726844209689, −4.11812793405040925842758448085, −3.21849600833024890890885982120, −2.31850101459127591054879767248, −1.25448862496615872642083988921, 1.25448862496615872642083988921, 2.31850101459127591054879767248, 3.21849600833024890890885982120, 4.11812793405040925842758448085, 5.15826309875877454726844209689, 5.79556690961774165079391688764, 7.34927148444945221696381934045, 7.78487293444204153358534125159, 8.167786562522512224681200743672, 9.255412632096169883997497117505

Graph of the ZZ-function along the critical line