↑

Properties

Label 2-2527-7.6-c0-0-4
Degree 22
Conductor 25272527
Sign 11
Analytic cond. 1.261131.26113
Root an. cond. 1.123001.12300
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.17·2-s + 0.381·4-s − 7-s + 0.726·8-s + 9-s + 1.61·11-s + 1.17·14-s − 1.23·16-s − 1.17·18-s − 1.90·22-s + 0.618·23-s + 25-s − 0.381·28-s − 1.90·29-s + 0.726·32-s + 0.381·36-s − 1.61·43-s + 0.618·44-s − 0.726·46-s + 49-s − 1.17·50-s + 1.17·53-s − 0.726·56-s + 2.23·58-s − 63-s + 0.381·64-s + 1.17·67-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.381·4-s − 7-s + 0.726·8-s + 9-s + 1.61·11-s + 1.17·14-s − 1.23·16-s − 1.17·18-s − 1.90·22-s + 0.618·23-s + 25-s − 0.381·28-s − 1.90·29-s + 0.726·32-s + 0.381·36-s − 1.61·43-s + 0.618·44-s − 0.726·46-s + 49-s − 1.17·50-s + 1.17·53-s − 0.726·56-s + 2.23·58-s − 63-s + 0.381·64-s + 1.17·67-s + ⋯

Functional equation

Λ(s)=(2527s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(2527s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25272527    =    7⋅1927 \cdot 19^{2}
Sign: 11
Analytic conductor: 1.261131.26113
Root analytic conductor: 1.123001.12300
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2527(1084,⋅)\chi_{2527} (1084, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2527, ( :0), 1)(2,\ 2527,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66292539170.6629253917
L(12)L(\frac12) ≈\approx 0.66292539170.6629253917
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)
bad7 1+T 1 + T
19 1 1
good2 1+1.17T+T2 1 + 1.17T + T^{2}
3 1−T2 1 - T^{2}
5 1−T2 1 - T^{2}
11 1−1.61T+T2 1 - 1.61T + T^{2}
13 1−T2 1 - T^{2}
17 1−T2 1 - T^{2}
23 1−0.618T+T2 1 - 0.618T + T^{2}
29 1+1.90T+T2 1 + 1.90T + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+1.61T+T2 1 + 1.61T + T^{2}
47 1−T2 1 - T^{2}
53 1−1.17T+T2 1 - 1.17T + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−1.17T+T2 1 - 1.17T + T^{2}
71 1+T2 1 + T^{2}
73 1−T2 1 - T^{2}
79 1−1.90T+T2 1 - 1.90T + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - T^{2}
show more
show less
   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

−9.203743631023791004363333834341, −8.621624232741404911220456351245, −7.54079475970765626826531841508, −6.89879094495389046916886732604, −6.47995354696416956065557806463, −5.17687471776181476460598889551, −4.12891555653162599963939373919, −3.48344006436650449533195192636, −1.94796406838159161512101365954, −0.970010285617325072201697358933, 0.970010285617325072201697358933, 1.94796406838159161512101365954, 3.48344006436650449533195192636, 4.12891555653162599963939373919, 5.17687471776181476460598889551, 6.47995354696416956065557806463, 6.89879094495389046916886732604, 7.54079475970765626826531841508, 8.621624232741404911220456351245, 9.203743631023791004363333834341

Graph of the ZZ-function along the critical line