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Properties

Label 2-1035-1.1-c1-0-12
Degree 22
Conductor 10351035
Sign 11
Analytic cond. 8.264518.26451
Root an. cond. 2.874802.87480
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  − 2.50·2-s + 4.25·4-s − 5-s + 3.78·7-s − 5.63·8-s + 2.50·10-s + 4.47·11-s + 5.13·13-s − 9.47·14-s + 5.59·16-s + 6.39·17-s − 7.47·19-s − 4.25·20-s − 11.1·22-s + 23-s + 25-s − 12.8·26-s + 16.1·28-s + 4.92·29-s − 1.25·31-s − 2.70·32-s − 15.9·34-s − 3.78·35-s + 6.54·37-s + 18.6·38-s + 5.63·40-s − 9.19·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 2.12·4-s − 0.447·5-s + 1.43·7-s − 1.99·8-s + 0.790·10-s + 1.34·11-s + 1.42·13-s − 2.53·14-s + 1.39·16-s + 1.55·17-s − 1.71·19-s − 0.951·20-s − 2.38·22-s + 0.208·23-s + 0.200·25-s − 2.51·26-s + 3.04·28-s + 0.914·29-s − 0.225·31-s − 0.478·32-s − 2.74·34-s − 0.640·35-s + 1.07·37-s + 3.03·38-s + 0.891·40-s − 1.43·41-s + ⋯

Functional equation

Λ(s)=(1035s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1035s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10351035    =    32⋅5⋅233^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 8.264518.26451
Root analytic conductor: 2.874802.87480
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1035, ( :1/2), 1)(2,\ 1035,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.91363700370.9136370037
L(12)L(\frac12) ≈\approx 0.91363700370.9136370037
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
23 1−T 1 - T
good2 1+2.50T+2T2 1 + 2.50T + 2T^{2}
7 1−3.78T+7T2 1 - 3.78T + 7T^{2}
11 1−4.47T+11T2 1 - 4.47T + 11T^{2}
13 1−5.13T+13T2 1 - 5.13T + 13T^{2}
17 1−6.39T+17T2 1 - 6.39T + 17T^{2}
19 1+7.47T+19T2 1 + 7.47T + 19T^{2}
29 1−4.92T+29T2 1 - 4.92T + 29T^{2}
31 1+1.25T+31T2 1 + 1.25T + 31T^{2}
37 1−6.54T+37T2 1 - 6.54T + 37T^{2}
41 1+9.19T+41T2 1 + 9.19T + 41T^{2}
43 1+3.17T+43T2 1 + 3.17T + 43T^{2}
47 1+6.71T+47T2 1 + 6.71T + 47T^{2}
53 1+11.3T+53T2 1 + 11.3T + 53T^{2}
59 1−3.65T+59T2 1 - 3.65T + 59T^{2}
61 1+8.98T+61T2 1 + 8.98T + 61T^{2}
67 1−15.1T+67T2 1 - 15.1T + 67T^{2}
71 1−8.33T+71T2 1 - 8.33T + 71T^{2}
73 1+1.13T+73T2 1 + 1.13T + 73T^{2}
79 1+15.7T+79T2 1 + 15.7T + 79T^{2}
83 1+3.06T+83T2 1 + 3.06T + 83T^{2}
89 1−13.4T+89T2 1 - 13.4T + 89T^{2}
97 1+4.42T+97T2 1 + 4.42T + 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

−9.864286071096951083481560796846, −8.850896580751320863055417489817, −8.322013241680387449519295015188, −7.927016130413064271171768237938, −6.79365851185933202677802106522, −6.12182354194648317471966197534, −4.62569122328417452265991821032, −3.46498437054268546461572285335, −1.77850276655201039451453895012, −1.08371708456086627154792319917, 1.08371708456086627154792319917, 1.77850276655201039451453895012, 3.46498437054268546461572285335, 4.62569122328417452265991821032, 6.12182354194648317471966197534, 6.79365851185933202677802106522, 7.927016130413064271171768237938, 8.322013241680387449519295015188, 8.850896580751320863055417489817, 9.864286071096951083481560796846

Graph of the ZZ-function along the critical line