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Properties

Label 2-1587-1.1-c1-0-26
Degree 22
Conductor 15871587
Sign 11
Analytic cond. 12.672212.6722
Root an. cond. 3.559813.55981
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  − 0.732·2-s − 3-s − 1.46·4-s + 0.732·5-s + 0.732·6-s + 3.73·7-s + 2.53·8-s + 9-s − 0.535·10-s + 4.73·11-s + 1.46·12-s + 13-s − 2.73·14-s − 0.732·15-s + 1.07·16-s + 3.26·17-s − 0.732·18-s + 5.46·19-s − 1.07·20-s − 3.73·21-s − 3.46·22-s − 2.53·24-s − 4.46·25-s − 0.732·26-s − 27-s − 5.46·28-s − 9.66·29-s + ⋯
L(s)  = 1  − 0.517·2-s − 0.577·3-s − 0.732·4-s + 0.327·5-s + 0.298·6-s + 1.41·7-s + 0.896·8-s + 0.333·9-s − 0.169·10-s + 1.42·11-s + 0.422·12-s + 0.277·13-s − 0.730·14-s − 0.189·15-s + 0.267·16-s + 0.792·17-s − 0.172·18-s + 1.25·19-s − 0.239·20-s − 0.814·21-s − 0.738·22-s − 0.517·24-s − 0.892·25-s − 0.143·26-s − 0.192·27-s − 1.03·28-s − 1.79·29-s + ⋯

Functional equation

Λ(s)=(1587s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1587s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15871587    =    3⋅2323 \cdot 23^{2}
Sign: 11
Analytic conductor: 12.672212.6722
Root analytic conductor: 3.559813.55981
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1587, ( :1/2), 1)(2,\ 1587,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3219502241.321950224
L(12)L(\frac12) ≈\approx 1.3219502241.321950224
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
23 1 1
good2 1+0.732T+2T2 1 + 0.732T + 2T^{2}
5 1−0.732T+5T2 1 - 0.732T + 5T^{2}
7 1−3.73T+7T2 1 - 3.73T + 7T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1−3.26T+17T2 1 - 3.26T + 17T^{2}
19 1−5.46T+19T2 1 - 5.46T + 19T^{2}
29 1+9.66T+29T2 1 + 9.66T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+3.19T+37T2 1 + 3.19T + 37T^{2}
41 1−9.46T+41T2 1 - 9.46T + 41T^{2}
43 1−9.73T+43T2 1 - 9.73T + 43T^{2}
47 1−2.19T+47T2 1 - 2.19T + 47T^{2}
53 1−4.92T+53T2 1 - 4.92T + 53T^{2}
59 1−7.66T+59T2 1 - 7.66T + 59T^{2}
61 1+7.19T+61T2 1 + 7.19T + 61T^{2}
67 1+13.7T+67T2 1 + 13.7T + 67T^{2}
71 1+6.19T+71T2 1 + 6.19T + 71T^{2}
73 1+6.92T+73T2 1 + 6.92T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+8.19T+89T2 1 + 8.19T + 89T^{2}
97 1+1.07T+97T2 1 + 1.07T + 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

−9.273722835226276255482589970341, −8.888471249866983017775379926225, −7.61985594474241822079543889054, −7.46668596655168341563425077685, −5.88921889767383060529179561493, −5.43750904875789304335230630180, −4.40472903806115616935550056854, −3.73095764450750418493617157698, −1.73504804923049686111992196544, −1.03062855264119002155536407222, 1.03062855264119002155536407222, 1.73504804923049686111992196544, 3.73095764450750418493617157698, 4.40472903806115616935550056854, 5.43750904875789304335230630180, 5.88921889767383060529179561493, 7.46668596655168341563425077685, 7.61985594474241822079543889054, 8.888471249866983017775379926225, 9.273722835226276255482589970341

Graph of the ZZ-function along the critical line