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Properties

Label 2-1058-1.1-c1-0-33
Degree 22
Conductor 10581058
Sign 11
Analytic cond. 8.448178.44817
Root an. cond. 2.906572.90657
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-s + 2.37·3-s + 4-s + 4.07·5-s + 2.37·6-s − 1.94·7-s + 8-s + 2.64·9-s + 4.07·10-s − 2.44·11-s + 2.37·12-s + 1.35·13-s − 1.94·14-s + 9.67·15-s + 16-s − 5.09·17-s + 2.64·18-s − 3.55·19-s + 4.07·20-s − 4.62·21-s − 2.44·22-s + 2.37·24-s + 11.5·25-s + 1.35·26-s − 0.846·27-s − 1.94·28-s − 4.40·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.37·3-s + 0.5·4-s + 1.82·5-s + 0.969·6-s − 0.735·7-s + 0.353·8-s + 0.881·9-s + 1.28·10-s − 0.738·11-s + 0.685·12-s + 0.376·13-s − 0.519·14-s + 2.49·15-s + 0.250·16-s − 1.23·17-s + 0.623·18-s − 0.815·19-s + 0.910·20-s − 1.00·21-s − 0.522·22-s + 0.484·24-s + 2.31·25-s + 0.266·26-s − 0.162·27-s − 0.367·28-s − 0.817·29-s + ⋯

Functional equation

Λ(s)=(1058s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1058s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10581058    =    2⋅2322 \cdot 23^{2}
Sign: 11
Analytic conductor: 8.448178.44817
Root analytic conductor: 2.906572.90657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1058, ( :1/2), 1)(2,\ 1058,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4905246454.490524645
L(12)L(\frac12) ≈\approx 4.4905246454.490524645
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)
bad2 1−T 1 - T
23 1 1
good3 1−2.37T+3T2 1 - 2.37T + 3T^{2}
5 1−4.07T+5T2 1 - 4.07T + 5T^{2}
7 1+1.94T+7T2 1 + 1.94T + 7T^{2}
11 1+2.44T+11T2 1 + 2.44T + 11T^{2}
13 1−1.35T+13T2 1 - 1.35T + 13T^{2}
17 1+5.09T+17T2 1 + 5.09T + 17T^{2}
19 1+3.55T+19T2 1 + 3.55T + 19T^{2}
29 1+4.40T+29T2 1 + 4.40T + 29T^{2}
31 1−3.46T+31T2 1 - 3.46T + 31T^{2}
37 1−1.94T+37T2 1 - 1.94T + 37T^{2}
41 1+5.57T+41T2 1 + 5.57T + 41T^{2}
43 1+1.63T+43T2 1 + 1.63T + 43T^{2}
47 1+2.29T+47T2 1 + 2.29T + 47T^{2}
53 1−8.84T+53T2 1 - 8.84T + 53T^{2}
59 1−14.3T+59T2 1 - 14.3T + 59T^{2}
61 1−5.96T+61T2 1 - 5.96T + 61T^{2}
67 1+8.83T+67T2 1 + 8.83T + 67T^{2}
71 1+13.9T+71T2 1 + 13.9T + 71T^{2}
73 1+4.92T+73T2 1 + 4.92T + 73T^{2}
79 1+7.06T+79T2 1 + 7.06T + 79T^{2}
83 1+4.96T+83T2 1 + 4.96T + 83T^{2}
89 1−16.0T+89T2 1 - 16.0T + 89T^{2}
97 1−8.59T+97T2 1 - 8.59T + 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

−9.986716168501808741683154296041, −8.969702442961656440461244141049, −8.560107715359345216361405801634, −7.23698678145384824922411227322, −6.39121347154709170576232238340, −5.72561776927678829548178726215, −4.61236107001189228301521193477, −3.38831153216863246886020998700, −2.49484509572493128102453217616, −1.94415903866258520207210824530, 1.94415903866258520207210824530, 2.49484509572493128102453217616, 3.38831153216863246886020998700, 4.61236107001189228301521193477, 5.72561776927678829548178726215, 6.39121347154709170576232238340, 7.23698678145384824922411227322, 8.560107715359345216361405801634, 8.969702442961656440461244141049, 9.986716168501808741683154296041

Graph of the ZZ-function along the critical line