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Properties

Label 2-2496-1.1-c1-0-6
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 19.930619.9306
Root an. cond. 4.464374.46437
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 − 2·7-s + 9-s + 4·11-s − 13-s − 6·17-s + 6·19-s + 2·21-s − 5·25-s − 27-s + 2·29-s + 6·31-s − 4·33-s − 10·37-s + 39-s + 8·41-s + 12·43-s − 12·47-s − 3·49-s + 6·51-s + 6·53-s − 6·57-s − 2·61-s − 2·63-s + 2·67-s + 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 0.949·71-s + 1.63·73-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 19.930619.9306
Root analytic conductor: 4.464374.46437
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :1/2), 1)(2,\ 2496,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2686003951.268600395
L(12)L(\frac12) ≈\approx 1.2686003951.268600395
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
13 1+T 1 + T
good5 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−4T+pT2 1 - 4 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

−9.220283570203917667028022793155, −8.137716429689322885874117903176, −7.16333848096912753102386607562, −6.57091938769509480942354725697, −5.98247083912518480972422452649, −4.97254580700680097900229302596, −4.13958008081477749813044275306, −3.29289095796707662296197413885, −2.06399639050261723146787386632, −0.73510269161810698907992625597, 0.73510269161810698907992625597, 2.06399639050261723146787386632, 3.29289095796707662296197413885, 4.13958008081477749813044275306, 4.97254580700680097900229302596, 5.98247083912518480972422452649, 6.57091938769509480942354725697, 7.16333848096912753102386607562, 8.137716429689322885874117903176, 9.220283570203917667028022793155

Graph of the ZZ-function along the critical line