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Properties

Label 2-612-68.67-c0-0-0
Degree 22
Conductor 612612
Sign 11
Analytic cond. 0.3054270.305427
Root an. cond. 0.5526550.552655
Motivic weight 00
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 + 8-s − 2·13-s + 16-s − 17-s + 25-s − 2·26-s + 32-s − 34-s − 49-s + 50-s − 2·52-s − 2·53-s + 64-s − 68-s + 2·89-s − 98-s + 100-s + 2·101-s − 2·104-s − 2·106-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 2·13-s + 16-s − 17-s + 25-s − 2·26-s + 32-s − 34-s − 49-s + 50-s − 2·52-s − 2·53-s + 64-s − 68-s + 2·89-s − 98-s + 100-s + 2·101-s − 2·104-s − 2·106-s + ⋯

Functional equation

Λ(s)=(612s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(612s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 612612    =    22⋅32⋅172^{2} \cdot 3^{2} \cdot 17
Sign: 11
Analytic conductor: 0.3054270.305427
Root analytic conductor: 0.5526550.552655
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ612(271,⋅)\chi_{612} (271, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 612, ( :0), 1)(2,\ 612,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5696166001.569616600
L(12)L(\frac12) ≈\approx 1.5696166001.569616600
L(1)L(1) 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
17 1+T 1 + T
good5 (1−T)(1+T) ( 1 - T )( 1 + T )
7 1+T2 1 + T^{2}
11 1+T2 1 + T^{2}
13 (1+T)2 ( 1 + T )^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 1+T2 1 + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T2 1 + T^{2}
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 (1+T)2 ( 1 + T )^{2}
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 1+T2 1 + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 1+T2 1 + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 (1−T)2 ( 1 - T )^{2}
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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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

−11.01361511296499662988051015116, −10.15934166306212921952565676537, −9.230068915079004819121392749342, −7.929900660265677048069781859922, −7.10461071371140693459800334853, −6.33572911611919867273565186629, −5.05357000362313766271212661191, −4.54815223255620235231881143477, −3.12333612951879445036966965769, −2.11496417031398947575557591105, 2.11496417031398947575557591105, 3.12333612951879445036966965769, 4.54815223255620235231881143477, 5.05357000362313766271212661191, 6.33572911611919867273565186629, 7.10461071371140693459800334853, 7.929900660265677048069781859922, 9.230068915079004819121392749342, 10.15934166306212921952565676537, 11.01361511296499662988051015116

Graph of the ZZ-function along the critical line