Properties

Label 2-1305-1.1-c1-0-0
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  − 0.414·2-s − 1.82·4-s − 5-s − 4.82·7-s + 1.58·8-s + 0.414·10-s − 0.828·11-s − 2·13-s + 1.99·14-s + 3·16-s − 2.82·17-s − 4.82·19-s + 1.82·20-s + 0.343·22-s + 3.17·23-s + 25-s + 0.828·26-s + 8.82·28-s − 29-s + 6.48·31-s − 4.41·32-s + 1.17·34-s + 4.82·35-s − 8.48·37-s + 1.99·38-s − 1.58·40-s + 6·41-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s − 0.447·5-s − 1.82·7-s + 0.560·8-s + 0.130·10-s − 0.249·11-s − 0.554·13-s + 0.534·14-s + 0.750·16-s − 0.685·17-s − 1.10·19-s + 0.408·20-s + 0.0731·22-s + 0.661·23-s + 0.200·25-s + 0.162·26-s + 1.66·28-s − 0.185·29-s + 1.16·31-s − 0.780·32-s + 0.200·34-s + 0.816·35-s − 1.39·37-s + 0.324·38-s − 0.250·40-s + 0.937·41-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

\(L(1)\) \(\approx\) \(0.4462992790\)
\(L(\frac12)\) \(\approx\) \(0.4462992790\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 4.82T + 19T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
31 \( 1 - 6.48T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 3.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 + 7.17T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.569735796860283104887902752783, −8.935221445141604486704813754802, −8.228094159143384664934506081671, −7.13012654704060293700489516664, −6.51560178460189214488763299907, −5.42031063076244516808654166364, −4.38427934121461106876595539207, −3.61131050761625910666601968286, −2.55173832700983347070171173196, −0.48854540853707181561886723604, 0.48854540853707181561886723604, 2.55173832700983347070171173196, 3.61131050761625910666601968286, 4.38427934121461106876595539207, 5.42031063076244516808654166364, 6.51560178460189214488763299907, 7.13012654704060293700489516664, 8.228094159143384664934506081671, 8.935221445141604486704813754802, 9.569735796860283104887902752783

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