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Properties

Label 2-68-68.67-c0-0-0
Degree 22
Conductor 6868
Sign 11
Analytic cond. 0.03393640.0339364
Root an. cond. 0.1842180.184218
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 − 9-s − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-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 − 9-s − 2·13-s + 16-s + 17-s + 18-s + 25-s + 2·26-s − 32-s − 34-s − 36-s − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯

Functional equation

Λ(s)=(68s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(68s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6868    =    22⋅172^{2} \cdot 17
Sign: 11
Analytic conductor: 0.03393640.0339364
Root analytic conductor: 0.1842180.184218
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ68(67,⋅)\chi_{68} (67, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 68, ( :0), 1)(2,\ 68,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.33278858670.3327885867
L(12)L(\frac12) ≈\approx 0.33278858670.3327885867
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
17 1−T 1 - T
good3 1+T2 1 + T^{2}
5 (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

−15.02449061344557603691883920300, −14.36118879225983677422230504881, −12.47090892005841483622787798176, −11.65059393536294409108661621379, −10.36498251692829322753123217545, −9.378387468381204143197364963903, −8.141436839216771787296709142564, −7.01903598371148299622349237377, −5.41265195424507937916978031210, −2.75562279580887633473300699062, 2.75562279580887633473300699062, 5.41265195424507937916978031210, 7.01903598371148299622349237377, 8.141436839216771787296709142564, 9.378387468381204143197364963903, 10.36498251692829322753123217545, 11.65059393536294409108661621379, 12.47090892005841483622787798176, 14.36118879225983677422230504881, 15.02449061344557603691883920300

Graph of the ZZ-function along the critical line