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Properties

Label 2-104-1.1-c1-0-2
Degree 22
Conductor 104104
Sign 11
Analytic cond. 0.8304440.830444
Root an. cond. 0.9112870.911287
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  + 2.56·3-s − 0.561·5-s − 2.56·7-s + 3.56·9-s − 5.12·11-s + 13-s − 1.43·15-s + 5.68·17-s + 5.12·19-s − 6.56·21-s − 8·23-s − 4.68·25-s + 1.43·27-s − 2·29-s + 4·31-s − 13.1·33-s + 1.43·35-s + 9.68·37-s + 2.56·39-s − 3.12·41-s + 5.43·43-s − 2.00·45-s − 0.315·47-s − 0.438·49-s + 14.5·51-s + 3.12·53-s + 2.87·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.251·5-s − 0.968·7-s + 1.18·9-s − 1.54·11-s + 0.277·13-s − 0.371·15-s + 1.37·17-s + 1.17·19-s − 1.43·21-s − 1.66·23-s − 0.936·25-s + 0.276·27-s − 0.371·29-s + 0.718·31-s − 2.28·33-s + 0.243·35-s + 1.59·37-s + 0.410·39-s − 0.487·41-s + 0.829·43-s − 0.298·45-s − 0.0459·47-s − 0.0626·49-s + 2.03·51-s + 0.428·53-s + 0.387·55-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: 11
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), 1)(2,\ 104,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3309952001.330995200
L(12)L(\frac12) ≈\approx 1.3309952001.330995200
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
13 1−T 1 - T
good3 1−2.56T+3T2 1 - 2.56T + 3T^{2}
5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1+2.56T+7T2 1 + 2.56T + 7T^{2}
11 1+5.12T+11T2 1 + 5.12T + 11T^{2}
17 1−5.68T+17T2 1 - 5.68T + 17T^{2}
19 1−5.12T+19T2 1 - 5.12T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−9.68T+37T2 1 - 9.68T + 37T^{2}
41 1+3.12T+41T2 1 + 3.12T + 41T^{2}
43 1−5.43T+43T2 1 - 5.43T + 43T^{2}
47 1+0.315T+47T2 1 + 0.315T + 47T^{2}
53 1−3.12T+53T2 1 - 3.12T + 53T^{2}
59 1−5.12T+59T2 1 - 5.12T + 59T^{2}
61 1−11.1T+61T2 1 - 11.1T + 61T^{2}
67 1+5.12T+67T2 1 + 5.12T + 67T^{2}
71 1+7.68T+71T2 1 + 7.68T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−2.24T+83T2 1 - 2.24T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+8.24T+97T2 1 + 8.24T + 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

−13.71753072328683946346817848159, −13.10195645373867280775639021405, −11.90591244910145441082720776956, −10.13223292975874914445498474652, −9.587407953010303947547664738248, −8.098504966471858535674901672523, −7.61495739064935920384504857216, −5.74788173249571342817361812177, −3.72784241969437990537506006035, −2.68088295258062156328134462169, 2.68088295258062156328134462169, 3.72784241969437990537506006035, 5.74788173249571342817361812177, 7.61495739064935920384504857216, 8.098504966471858535674901672523, 9.587407953010303947547664738248, 10.13223292975874914445498474652, 11.90591244910145441082720776956, 13.10195645373867280775639021405, 13.71753072328683946346817848159

Graph of the ZZ-function along the critical line