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Properties

Label 2-1560-1.1-c1-0-15
Degree 22
Conductor 15601560
Sign 11
Analytic cond. 12.456612.4566
Root an. cond. 3.529393.52939
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  + 3-s + 5-s + 4·7-s + 9-s + 13-s + 15-s + 2·17-s + 4·21-s + 25-s + 27-s − 2·29-s − 4·31-s + 4·35-s + 6·37-s + 39-s − 6·41-s + 4·43-s + 45-s − 4·47-s + 9·49-s + 2·51-s − 10·53-s − 2·61-s + 4·63-s + 65-s + 8·67-s + 4·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.485·17-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.280·51-s − 1.37·53-s − 0.256·61-s + 0.503·63-s + 0.124·65-s + 0.977·67-s + 0.474·71-s + ⋯

Functional equation

Λ(s)=(1560s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1560s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15601560    =    23⋅3⋅5⋅132^{3} \cdot 3 \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 12.456612.4566
Root analytic conductor: 3.529393.52939
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1560, ( :1/2), 1)(2,\ 1560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7903758082.790375808
L(12)L(\frac12) ≈\approx 2.7903758082.790375808
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
3 1−T 1 - T
5 1−T 1 - T
13 1−T 1 - T
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−9.361922981370355726520537378237, −8.560865380512469096549492525929, −7.937701919404004482623022118064, −7.25810834528999964875509262651, −6.12257508120125303222030801610, −5.22066779295482831433366349716, −4.46333026419845538385546929457, −3.38719073586029414138569537242, −2.17760885066242309825446481805, −1.33526849821080802501597040286, 1.33526849821080802501597040286, 2.17760885066242309825446481805, 3.38719073586029414138569537242, 4.46333026419845538385546929457, 5.22066779295482831433366349716, 6.12257508120125303222030801610, 7.25810834528999964875509262651, 7.937701919404004482623022118064, 8.560865380512469096549492525929, 9.361922981370355726520537378237

Graph of the ZZ-function along the critical line