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Properties

Label 2-1305-1.1-c1-0-10
Degree 22
Conductor 13051305
Sign 11
Analytic cond. 10.420410.4204
Root an. cond. 3.228073.22807
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  − 2.66·2-s + 5.07·4-s + 5-s + 4.68·7-s − 8.19·8-s − 2.66·10-s − 3.50·11-s − 4.56·13-s − 12.4·14-s + 11.6·16-s + 6.56·17-s − 4.10·19-s + 5.07·20-s + 9.32·22-s + 6.92·23-s + 25-s + 12.1·26-s + 23.7·28-s − 29-s + 8.10·31-s − 14.5·32-s − 17.4·34-s + 4.68·35-s + 4.27·37-s + 10.9·38-s − 8.19·40-s − 5.10·41-s + ⋯
L(s)  = 1  − 1.88·2-s + 2.53·4-s + 0.447·5-s + 1.76·7-s − 2.89·8-s − 0.841·10-s − 1.05·11-s − 1.26·13-s − 3.32·14-s + 2.91·16-s + 1.59·17-s − 0.942·19-s + 1.13·20-s + 1.98·22-s + 1.44·23-s + 0.200·25-s + 2.38·26-s + 4.49·28-s − 0.185·29-s + 1.45·31-s − 2.57·32-s − 2.99·34-s + 0.791·35-s + 0.702·37-s + 1.77·38-s − 1.29·40-s − 0.797·41-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 10.420410.4204
Root analytic conductor: 3.228073.22807
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1305, ( :1/2), 1)(2,\ 1305,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88550677100.8855067710
L(12)L(\frac12) ≈\approx 0.88550677100.8855067710
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−T 1 - T
29 1+T 1 + T
good2 1+2.66T+2T2 1 + 2.66T + 2T^{2}
7 1−4.68T+7T2 1 - 4.68T + 7T^{2}
11 1+3.50T+11T2 1 + 3.50T + 11T^{2}
13 1+4.56T+13T2 1 + 4.56T + 13T^{2}
17 1−6.56T+17T2 1 - 6.56T + 17T^{2}
19 1+4.10T+19T2 1 + 4.10T + 19T^{2}
23 1−6.92T+23T2 1 - 6.92T + 23T^{2}
31 1−8.10T+31T2 1 - 8.10T + 31T^{2}
37 1−4.27T+37T2 1 - 4.27T + 37T^{2}
41 1+5.10T+41T2 1 + 5.10T + 41T^{2}
43 1+8.47T+43T2 1 + 8.47T + 43T^{2}
47 1−5.41T+47T2 1 - 5.41T + 47T^{2}
53 1−4.20T+53T2 1 - 4.20T + 53T^{2}
59 1−5.36T+59T2 1 - 5.36T + 59T^{2}
61 1−1.27T+61T2 1 - 1.27T + 61T^{2}
67 1−9.96T+67T2 1 - 9.96T + 67T^{2}
71 1+12.2T+71T2 1 + 12.2T + 71T^{2}
73 1−8.37T+73T2 1 - 8.37T + 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1−14.6T+83T2 1 - 14.6T + 83T^{2}
89 1−2.20T+89T2 1 - 2.20T + 89T^{2}
97 1−5.46T+97T2 1 - 5.46T + 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

−9.730955696061759256548690302413, −8.691532990679175642361826726721, −8.104743122307452318525578101850, −7.60314183237129379068350510049, −6.82259209356041568925178562274, −5.52780307981793170979619684669, −4.85244140875330066387011520809, −2.80572240898260933064835461589, −2.02802529977135241119668376759, −0.941004506408326481606102938697, 0.941004506408326481606102938697, 2.02802529977135241119668376759, 2.80572240898260933064835461589, 4.85244140875330066387011520809, 5.52780307981793170979619684669, 6.82259209356041568925178562274, 7.60314183237129379068350510049, 8.104743122307452318525578101850, 8.691532990679175642361826726721, 9.730955696061759256548690302413

Graph of the ZZ-function along the critical line