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Properties

Label 2-195-1.1-c5-0-24
Degree 22
Conductor 195195
Sign −1-1
Analytic cond. 31.274831.2748
Root an. cond. 5.592395.59239
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.14·2-s + 9·3-s + 51.6·4-s − 25·5-s − 82.3·6-s + 24.2·7-s − 180.·8-s + 81·9-s + 228.·10-s − 126.·11-s + 465.·12-s + 169·13-s − 221.·14-s − 225·15-s − 6.48·16-s − 912.·17-s − 740.·18-s + 507.·19-s − 1.29e3·20-s + 217.·21-s + 1.15e3·22-s − 1.47e3·23-s − 1.62e3·24-s + 625·25-s − 1.54e3·26-s + 729·27-s + 1.25e3·28-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.577·3-s + 1.61·4-s − 0.447·5-s − 0.933·6-s + 0.186·7-s − 0.994·8-s + 0.333·9-s + 0.723·10-s − 0.315·11-s + 0.932·12-s + 0.277·13-s − 0.301·14-s − 0.258·15-s − 0.00633·16-s − 0.766·17-s − 0.539·18-s + 0.322·19-s − 0.722·20-s + 0.107·21-s + 0.509·22-s − 0.581·23-s − 0.574·24-s + 0.200·25-s − 0.448·26-s + 0.192·27-s + 0.301·28-s + ⋯

Functional equation

Λ(s)=(195s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(195s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 195195    =    3⋅5⋅133 \cdot 5 \cdot 13
Sign: −1-1
Analytic conductor: 31.274831.2748
Root analytic conductor: 5.592395.59239
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 195, ( :5/2), −1)(2,\ 195,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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)
bad3 1−9T 1 - 9T
5 1+25T 1 + 25T
13 1−169T 1 - 169T
good2 1+9.14T+32T2 1 + 9.14T + 32T^{2}
7 1−24.2T+1.68e4T2 1 - 24.2T + 1.68e4T^{2}
11 1+126.T+1.61e5T2 1 + 126.T + 1.61e5T^{2}
17 1+912.T+1.41e6T2 1 + 912.T + 1.41e6T^{2}
19 1−507.T+2.47e6T2 1 - 507.T + 2.47e6T^{2}
23 1+1.47e3T+6.43e6T2 1 + 1.47e3T + 6.43e6T^{2}
29 1+315.T+2.05e7T2 1 + 315.T + 2.05e7T^{2}
31 1−3.16e3T+2.86e7T2 1 - 3.16e3T + 2.86e7T^{2}
37 1−1.46e4T+6.93e7T2 1 - 1.46e4T + 6.93e7T^{2}
41 1+3.92e3T+1.15e8T2 1 + 3.92e3T + 1.15e8T^{2}
43 1+1.15e3T+1.47e8T2 1 + 1.15e3T + 1.47e8T^{2}
47 1−4.33e3T+2.29e8T2 1 - 4.33e3T + 2.29e8T^{2}
53 1+2.54e4T+4.18e8T2 1 + 2.54e4T + 4.18e8T^{2}
59 1+3.23e4T+7.14e8T2 1 + 3.23e4T + 7.14e8T^{2}
61 1+2.87e4T+8.44e8T2 1 + 2.87e4T + 8.44e8T^{2}
67 1−5.11e3T+1.35e9T2 1 - 5.11e3T + 1.35e9T^{2}
71 1+4.82e4T+1.80e9T2 1 + 4.82e4T + 1.80e9T^{2}
73 1−9.88e3T+2.07e9T2 1 - 9.88e3T + 2.07e9T^{2}
79 1+6.96e4T+3.07e9T2 1 + 6.96e4T + 3.07e9T^{2}
83 1+3.67e4T+3.93e9T2 1 + 3.67e4T + 3.93e9T^{2}
89 1+4.83e4T+5.58e9T2 1 + 4.83e4T + 5.58e9T^{2}
97 1+8.05e4T+8.58e9T2 1 + 8.05e4T + 8.58e9T^{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

−10.89225509197496627266135567447, −9.910348416505816484392011584798, −9.056503678323827864141931705861, −8.162551415726364481742470069843, −7.56024605628931994926660008009, −6.37099027776451040979764123981, −4.44424678719721967154390018106, −2.78228297888674052298435819864, −1.44711984119799416649266218933, 0, 1.44711984119799416649266218933, 2.78228297888674052298435819864, 4.44424678719721967154390018106, 6.37099027776451040979764123981, 7.56024605628931994926660008009, 8.162551415726364481742470069843, 9.056503678323827864141931705861, 9.910348416505816484392011584798, 10.89225509197496627266135567447

Graph of the ZZ-function along the critical line