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Properties

Label 2-507-1.1-c1-0-17
Degree 22
Conductor 507507
Sign 11
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + 2.35·2-s − 3-s + 3.55·4-s + 3.69·5-s − 2.35·6-s − 0.801·7-s + 3.66·8-s + 9-s + 8.70·10-s − 2.85·11-s − 3.55·12-s − 1.89·14-s − 3.69·15-s + 1.52·16-s + 2.93·17-s + 2.35·18-s − 2.44·19-s + 13.1·20-s + 0.801·21-s − 6.71·22-s − 7.78·23-s − 3.66·24-s + 8.63·25-s − 27-s − 2.85·28-s + 3.85·29-s − 8.70·30-s + ⋯
L(s)  = 1  + 1.66·2-s − 0.577·3-s + 1.77·4-s + 1.65·5-s − 0.962·6-s − 0.303·7-s + 1.29·8-s + 0.333·9-s + 2.75·10-s − 0.859·11-s − 1.02·12-s − 0.505·14-s − 0.953·15-s + 0.381·16-s + 0.712·17-s + 0.555·18-s − 0.560·19-s + 2.93·20-s + 0.174·21-s − 1.43·22-s − 1.62·23-s − 0.748·24-s + 1.72·25-s − 0.192·27-s − 0.538·28-s + 0.715·29-s − 1.58·30-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 11
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 1)(2,\ 507,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5154634223.515463422
L(12)L(\frac12) ≈\approx 3.5154634223.515463422
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
13 1 1
good2 1−2.35T+2T2 1 - 2.35T + 2T^{2}
5 1−3.69T+5T2 1 - 3.69T + 5T^{2}
7 1+0.801T+7T2 1 + 0.801T + 7T^{2}
11 1+2.85T+11T2 1 + 2.85T + 11T^{2}
17 1−2.93T+17T2 1 - 2.93T + 17T^{2}
19 1+2.44T+19T2 1 + 2.44T + 19T^{2}
23 1+7.78T+23T2 1 + 7.78T + 23T^{2}
29 1−3.85T+29T2 1 - 3.85T + 29T^{2}
31 1−2.34T+31T2 1 - 2.34T + 31T^{2}
37 1−7.44T+37T2 1 - 7.44T + 37T^{2}
41 1+0.850T+41T2 1 + 0.850T + 41T^{2}
43 1+1.61T+43T2 1 + 1.61T + 43T^{2}
47 1−2.44T+47T2 1 - 2.44T + 47T^{2}
53 1+9.96T+53T2 1 + 9.96T + 53T^{2}
59 1+5.38T+59T2 1 + 5.38T + 59T^{2}
61 1+13.2T+61T2 1 + 13.2T + 61T^{2}
67 1−14.3T+67T2 1 - 14.3T + 67T^{2}
71 1−8.12T+71T2 1 - 8.12T + 71T^{2}
73 1+11.8T+73T2 1 + 11.8T + 73T^{2}
79 1−5.40T+79T2 1 - 5.40T + 79T^{2}
83 1−7.04T+83T2 1 - 7.04T + 83T^{2}
89 1−1.13T+89T2 1 - 1.13T + 89T^{2}
97 1+5.94T+97T2 1 + 5.94T + 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

−11.02428000056674997789312650067, −10.22851001352422597459392482860, −9.541523968179841066512171474452, −7.931350372598345093761783796969, −6.49836467154880290748927830507, −6.08007829497662988317020565119, −5.34355285688175209912160995369, −4.47555183432868305616718971206, −3.00726401005179861311751062407, −1.96556914197670966460621889314, 1.96556914197670966460621889314, 3.00726401005179861311751062407, 4.47555183432868305616718971206, 5.34355285688175209912160995369, 6.08007829497662988317020565119, 6.49836467154880290748927830507, 7.931350372598345093761783796969, 9.541523968179841066512171474452, 10.22851001352422597459392482860, 11.02428000056674997789312650067

Graph of the ZZ-function along the critical line