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Properties

Label 2-378-1.1-c1-0-1
Degree 22
Conductor 378378
Sign 11
Analytic cond. 3.018343.01834
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 5·13-s − 14-s + 16-s − 3·17-s + 2·19-s + 9·23-s − 5·25-s − 5·26-s + 28-s + 3·29-s + 5·31-s − 32-s + 3·34-s + 2·37-s − 2·38-s + 6·41-s − 43-s − 9·46-s + 6·47-s + 49-s + 5·50-s + 5·52-s − 3·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 1.87·23-s − 25-s − 0.980·26-s + 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.324·38-s + 0.937·41-s − 0.152·43-s − 1.32·46-s + 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.693·52-s − 0.412·53-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 11
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 1)(2,\ 378,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0818924741.081892474
L(12)L(\frac12) ≈\approx 1.0818924741.081892474
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+T 1 + T
3 1 1
7 1−T 1 - T
good5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−9T+pT2 1 - 9 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+13T+pT2 1 + 13 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−8T+pT2 1 - 8 T + p T^{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

−11.15664487403838869778857539841, −10.57904738545541172722863200266, −9.358886306281919204370388916695, −8.682143010158560978147219777821, −7.77295226765590417072124632150, −6.72994634489814968752118634163, −5.74408802320171077841105153966, −4.35441080626390222166206464206, −2.89963569250000254763681159752, −1.25841763926025585348683837537, 1.25841763926025585348683837537, 2.89963569250000254763681159752, 4.35441080626390222166206464206, 5.74408802320171077841105153966, 6.72994634489814968752118634163, 7.77295226765590417072124632150, 8.682143010158560978147219777821, 9.358886306281919204370388916695, 10.57904738545541172722863200266, 11.15664487403838869778857539841

Graph of the ZZ-function along the critical line