Properties

Label 2-1305-1.1-c1-0-2
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  − 0.431·2-s − 1.81·4-s − 5-s − 2.42·7-s + 1.64·8-s + 0.431·10-s + 0.00755·11-s − 2.73·13-s + 1.04·14-s + 2.91·16-s − 4.73·17-s + 2.74·19-s + 1.81·20-s − 0.00325·22-s − 3.68·23-s + 25-s + 1.17·26-s + 4.39·28-s + 29-s + 1.25·31-s − 4.55·32-s + 2.04·34-s + 2.42·35-s + 6.60·37-s − 1.18·38-s − 1.64·40-s − 0.160·41-s + ⋯
L(s)  = 1  − 0.305·2-s − 0.906·4-s − 0.447·5-s − 0.916·7-s + 0.581·8-s + 0.136·10-s + 0.00227·11-s − 0.757·13-s + 0.279·14-s + 0.729·16-s − 1.14·17-s + 0.629·19-s + 0.405·20-s − 0.000694·22-s − 0.769·23-s + 0.200·25-s + 0.231·26-s + 0.831·28-s + 0.185·29-s + 0.225·31-s − 0.804·32-s + 0.350·34-s + 0.409·35-s + 1.08·37-s − 0.192·38-s − 0.260·40-s − 0.0250·41-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

\(L(1)\) \(\approx\) \(0.6473634824\)
\(L(\frac12)\) \(\approx\) \(0.6473634824\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.431T + 2T^{2} \)
7 \( 1 + 2.42T + 7T^{2} \)
11 \( 1 - 0.00755T + 11T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 - 6.60T + 37T^{2} \)
41 \( 1 + 0.160T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 4.69T + 67T^{2} \)
71 \( 1 + 1.48T + 71T^{2} \)
73 \( 1 - 2.75T + 73T^{2} \)
79 \( 1 - 2.72T + 79T^{2} \)
83 \( 1 - 7.81T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 + 7.02T + 97T^{2} \)
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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

−9.683230084821019193660802014871, −8.907144795471558152403434389723, −8.123241474959971546565834492580, −7.31463646524735052173917427989, −6.43711679445481108946764567681, −5.36177741095526168758754142875, −4.42450979705355023791013292884, −3.67434634179096979262001656158, −2.45050030337405982599065202299, −0.60210035958858252656005897022, 0.60210035958858252656005897022, 2.45050030337405982599065202299, 3.67434634179096979262001656158, 4.42450979705355023791013292884, 5.36177741095526168758754142875, 6.43711679445481108946764567681, 7.31463646524735052173917427989, 8.123241474959971546565834492580, 8.907144795471558152403434389723, 9.683230084821019193660802014871

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