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Properties

Label 2-4304-1.1-c1-0-107
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
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  − 1.34·3-s + 3.90·5-s + 0.487·7-s − 1.19·9-s − 4.04·11-s + 3.53·13-s − 5.24·15-s − 6.37·17-s − 0.975·19-s − 0.655·21-s − 0.750·23-s + 10.2·25-s + 5.63·27-s − 4.08·29-s + 7.86·31-s + 5.44·33-s + 1.90·35-s − 4.22·37-s − 4.75·39-s − 2.07·41-s − 10.7·43-s − 4.66·45-s − 9.31·47-s − 6.76·49-s + 8.56·51-s + 11.1·53-s − 15.8·55-s + ⋯
L(s)  = 1  − 0.776·3-s + 1.74·5-s + 0.184·7-s − 0.397·9-s − 1.22·11-s + 0.981·13-s − 1.35·15-s − 1.54·17-s − 0.223·19-s − 0.143·21-s − 0.156·23-s + 2.05·25-s + 1.08·27-s − 0.757·29-s + 1.41·31-s + 0.947·33-s + 0.322·35-s − 0.694·37-s − 0.761·39-s − 0.323·41-s − 1.63·43-s − 0.694·45-s − 1.35·47-s − 0.965·49-s + 1.19·51-s + 1.52·53-s − 2.13·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :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)
bad2 1 1
269 1−T 1 - T
good3 1+1.34T+3T2 1 + 1.34T + 3T^{2}
5 1−3.90T+5T2 1 - 3.90T + 5T^{2}
7 1−0.487T+7T2 1 - 0.487T + 7T^{2}
11 1+4.04T+11T2 1 + 4.04T + 11T^{2}
13 1−3.53T+13T2 1 - 3.53T + 13T^{2}
17 1+6.37T+17T2 1 + 6.37T + 17T^{2}
19 1+0.975T+19T2 1 + 0.975T + 19T^{2}
23 1+0.750T+23T2 1 + 0.750T + 23T^{2}
29 1+4.08T+29T2 1 + 4.08T + 29T^{2}
31 1−7.86T+31T2 1 - 7.86T + 31T^{2}
37 1+4.22T+37T2 1 + 4.22T + 37T^{2}
41 1+2.07T+41T2 1 + 2.07T + 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1+9.31T+47T2 1 + 9.31T + 47T^{2}
53 1−11.1T+53T2 1 - 11.1T + 53T^{2}
59 1−2.98T+59T2 1 - 2.98T + 59T^{2}
61 1+7.53T+61T2 1 + 7.53T + 61T^{2}
67 1+3.51T+67T2 1 + 3.51T + 67T^{2}
71 1+14.1T+71T2 1 + 14.1T + 71T^{2}
73 1+9.75T+73T2 1 + 9.75T + 73T^{2}
79 1+2.29T+79T2 1 + 2.29T + 79T^{2}
83 1−17.1T+83T2 1 - 17.1T + 83T^{2}
89 1−1.86T+89T2 1 - 1.86T + 89T^{2}
97 1+9.63T+97T2 1 + 9.63T + 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

−8.293681672390597665803250741591, −6.95902862183388946670158939829, −6.32859393734845081222928972925, −5.88625135694040973037667429887, −5.15446269506707991558664823736, −4.63409495869031594267188865522, −3.15273637781708652618152298017, −2.31402949087458923211614839918, −1.50539721523385660141825486145, 0, 1.50539721523385660141825486145, 2.31402949087458923211614839918, 3.15273637781708652618152298017, 4.63409495869031594267188865522, 5.15446269506707991558664823736, 5.88625135694040973037667429887, 6.32859393734845081222928972925, 6.95902862183388946670158939829, 8.293681672390597665803250741591

Graph of the ZZ-function along the critical line