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Properties

Label 2-2475-1.1-c1-0-48
Degree 22
Conductor 24752475
Sign 11
Analytic cond. 19.762919.7629
Root an. cond. 4.445554.44555
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·2-s + 5.15·4-s − 2.80·7-s + 8.44·8-s + 11-s + 5.11·13-s − 7.50·14-s + 12.2·16-s + 4.54·17-s − 4.57·19-s + 2.67·22-s + 4·23-s + 13.6·26-s − 14.4·28-s + 2.38·29-s − 0.962·31-s + 15.9·32-s + 12.1·34-s − 1.61·37-s − 12.2·38-s + 2.38·41-s − 2.80·43-s + 5.15·44-s + 10.7·46-s − 4.31·47-s + 0.873·49-s + 26.3·52-s + ⋯
L(s)  = 1  + 1.89·2-s + 2.57·4-s − 1.06·7-s + 2.98·8-s + 0.301·11-s + 1.41·13-s − 2.00·14-s + 3.06·16-s + 1.10·17-s − 1.04·19-s + 0.570·22-s + 0.834·23-s + 2.68·26-s − 2.73·28-s + 0.443·29-s − 0.172·31-s + 2.81·32-s + 2.08·34-s − 0.265·37-s − 1.98·38-s + 0.372·41-s − 0.427·43-s + 0.777·44-s + 1.57·46-s − 0.629·47-s + 0.124·49-s + 3.66·52-s + ⋯

Functional equation

Λ(s)=(2475s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2475s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24752475    =    32⋅52⋅113^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 19.762919.7629
Root analytic conductor: 4.445554.44555
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2475, ( :1/2), 1)(2,\ 2475,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.1413999166.141399916
L(12)L(\frac12) ≈\approx 6.1413999166.141399916
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)
bad3 1 1
5 1 1
11 1−T 1 - T
good2 1−2.67T+2T2 1 - 2.67T + 2T^{2}
7 1+2.80T+7T2 1 + 2.80T + 7T^{2}
13 1−5.11T+13T2 1 - 5.11T + 13T^{2}
17 1−4.54T+17T2 1 - 4.54T + 17T^{2}
19 1+4.57T+19T2 1 + 4.57T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1−2.38T+29T2 1 - 2.38T + 29T^{2}
31 1+0.962T+31T2 1 + 0.962T + 31T^{2}
37 1+1.61T+37T2 1 + 1.61T + 37T^{2}
41 1−2.38T+41T2 1 - 2.38T + 41T^{2}
43 1+2.80T+43T2 1 + 2.80T + 43T^{2}
47 1+4.31T+47T2 1 + 4.31T + 47T^{2}
53 1+6.57T+53T2 1 + 6.57T + 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1−7.92T+61T2 1 - 7.92T + 61T^{2}
67 1+10.7T+67T2 1 + 10.7T + 67T^{2}
71 1−7.35T+71T2 1 - 7.35T + 71T^{2}
73 1+6.41T+73T2 1 + 6.41T + 73T^{2}
79 1−1.35T+79T2 1 - 1.35T + 79T^{2}
83 1+0.806T+83T2 1 + 0.806T + 83T^{2}
89 1−2.96T+89T2 1 - 2.96T + 89T^{2}
97 1−9.92T+97T2 1 - 9.92T + 97T^{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

−8.857846498097446685269635268276, −7.921750100089480391720263638626, −6.84540100021831471455704906482, −6.43844244304904878367628584796, −5.79306765203612391501947305796, −4.98202639714813758689288300376, −3.92729288341920008085240274453, −3.47997674795130572765919602868, −2.66465030038503452559749227258, −1.36397059200448912322557757740, 1.36397059200448912322557757740, 2.66465030038503452559749227258, 3.47997674795130572765919602868, 3.92729288341920008085240274453, 4.98202639714813758689288300376, 5.79306765203612391501947305796, 6.43844244304904878367628584796, 6.84540100021831471455704906482, 7.921750100089480391720263638626, 8.857846498097446685269635268276

Graph of the ZZ-function along the critical line