Properties

Label 2-1305-1.1-c1-0-5
Degree $2$
Conductor $1305$
Sign $1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.43·2-s + 0.0686·4-s + 5-s − 2.74·7-s + 2.77·8-s − 1.43·10-s − 2.74·11-s − 5.14·13-s + 3.94·14-s − 4.13·16-s + 3.72·17-s − 0.404·19-s + 0.0686·20-s + 3.94·22-s + 5.45·23-s + 25-s + 7.40·26-s − 0.188·28-s + 29-s + 1.45·31-s + 0.388·32-s − 5.36·34-s − 2.74·35-s + 6.76·37-s + 0.581·38-s + 2.77·40-s − 9.78·41-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.0343·4-s + 0.447·5-s − 1.03·7-s + 0.982·8-s − 0.454·10-s − 0.827·11-s − 1.42·13-s + 1.05·14-s − 1.03·16-s + 0.904·17-s − 0.0927·19-s + 0.0153·20-s + 0.841·22-s + 1.13·23-s + 0.200·25-s + 1.45·26-s − 0.0355·28-s + 0.185·29-s + 0.261·31-s + 0.0686·32-s − 0.919·34-s − 0.463·35-s + 1.11·37-s + 0.0943·38-s + 0.439·40-s − 1.52·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6464349571\)
\(L(\frac12)\) \(\approx\) \(0.6464349571\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 + 1.43T + 2T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 + 2.74T + 11T^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
17 \( 1 - 3.72T + 17T^{2} \)
19 \( 1 + 0.404T + 19T^{2} \)
23 \( 1 - 5.45T + 23T^{2} \)
31 \( 1 - 1.45T + 31T^{2} \)
37 \( 1 - 6.76T + 37T^{2} \)
41 \( 1 + 9.78T + 41T^{2} \)
43 \( 1 - 4.43T + 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 - 6.43T + 53T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 1.62T + 83T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 - 7.82T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.740578729232786388127976184331, −9.037040314977176693180496465746, −8.076118133452586258150504044492, −7.36970899871218492111223073568, −6.61352613631778993665096036846, −5.41505195687278823753276083115, −4.70777471881903065136635914776, −3.25606953653899693689065639521, −2.26388666047203924476833680422, −0.67013277183930208911141896232, 0.67013277183930208911141896232, 2.26388666047203924476833680422, 3.25606953653899693689065639521, 4.70777471881903065136635914776, 5.41505195687278823753276083115, 6.61352613631778993665096036846, 7.36970899871218492111223073568, 8.076118133452586258150504044492, 9.037040314977176693180496465746, 9.740578729232786388127976184331

Graph of the $Z$-function along the critical line