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Properties

Label 2-525-1.1-c1-0-6
Degree 22
Conductor 525525
Sign 11
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
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  − 0.193·2-s + 3-s − 1.96·4-s − 0.193·6-s + 7-s + 0.768·8-s + 9-s + 2·11-s − 1.96·12-s − 1.35·13-s − 0.193·14-s + 3.77·16-s + 3.35·17-s − 0.193·18-s + 5.35·19-s + 21-s − 0.387·22-s − 4.96·23-s + 0.768·24-s + 0.261·26-s + 27-s − 1.96·28-s + 7.92·29-s + 4.57·31-s − 2.26·32-s + 2·33-s − 0.649·34-s + ⋯
L(s)  = 1  − 0.137·2-s + 0.577·3-s − 0.981·4-s − 0.0791·6-s + 0.377·7-s + 0.271·8-s + 0.333·9-s + 0.603·11-s − 0.566·12-s − 0.374·13-s − 0.0518·14-s + 0.943·16-s + 0.812·17-s − 0.0457·18-s + 1.22·19-s + 0.218·21-s − 0.0826·22-s − 1.03·23-s + 0.156·24-s + 0.0513·26-s + 0.192·27-s − 0.370·28-s + 1.47·29-s + 0.821·31-s − 0.401·32-s + 0.348·33-s − 0.111·34-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), 1)(2,\ 525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4545834861.454583486
L(12)L(\frac12) ≈\approx 1.4545834861.454583486
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−T 1 - T
5 1 1
7 1−T 1 - T
good2 1+0.193T+2T2 1 + 0.193T + 2T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+1.35T+13T2 1 + 1.35T + 13T^{2}
17 1−3.35T+17T2 1 - 3.35T + 17T^{2}
19 1−5.35T+19T2 1 - 5.35T + 19T^{2}
23 1+4.96T+23T2 1 + 4.96T + 23T^{2}
29 1−7.92T+29T2 1 - 7.92T + 29T^{2}
31 1−4.57T+31T2 1 - 4.57T + 31T^{2}
37 1−0.775T+37T2 1 - 0.775T + 37T^{2}
41 1−3.73T+41T2 1 - 3.73T + 41T^{2}
43 1−12.6T+43T2 1 - 12.6T + 43T^{2}
47 1+9.92T+47T2 1 + 9.92T + 47T^{2}
53 1+8.57T+53T2 1 + 8.57T + 53T^{2}
59 1+8.62T+59T2 1 + 8.62T + 59T^{2}
61 1+8.70T+61T2 1 + 8.70T + 61T^{2}
67 1+9.92T+67T2 1 + 9.92T + 67T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−9.35T+73T2 1 - 9.35T + 73T^{2}
79 1−10.7T+79T2 1 - 10.7T + 79T^{2}
83 1+3.22T+83T2 1 + 3.22T + 83T^{2}
89 1−1.03T+89T2 1 - 1.03T + 89T^{2}
97 1−18.4T+97T2 1 - 18.4T + 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

−10.61797598792780120314623660474, −9.697774530240902187425587204845, −9.240701171692835591998690599415, −8.095944194541780680855303610191, −7.68234739127162575410322514300, −6.22924475067329073486711598694, −5.02522262844140395814578506502, −4.15816000723335585044553795505, −3.00963324380550823526572834362, −1.22392924948101114786523316269, 1.22392924948101114786523316269, 3.00963324380550823526572834362, 4.15816000723335585044553795505, 5.02522262844140395814578506502, 6.22924475067329073486711598694, 7.68234739127162575410322514300, 8.095944194541780680855303610191, 9.240701171692835591998690599415, 9.697774530240902187425587204845, 10.61797598792780120314623660474

Graph of the ZZ-function along the critical line