Properties

Label 2-1305-1.1-c1-0-41
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  + 2.75·2-s + 5.59·4-s + 5-s + 0.393·7-s + 9.92·8-s + 2.75·10-s + 0.393·11-s − 2.56·13-s + 1.08·14-s + 16.1·16-s − 2.07·17-s − 0.958·19-s + 5.59·20-s + 1.08·22-s − 6.15·23-s + 25-s − 7.07·26-s + 2.20·28-s + 29-s − 10.1·31-s + 24.6·32-s − 5.72·34-s + 0.393·35-s − 7.34·37-s − 2.64·38-s + 9.92·40-s + 1.65·41-s + ⋯
L(s)  = 1  + 1.94·2-s + 2.79·4-s + 0.447·5-s + 0.148·7-s + 3.50·8-s + 0.871·10-s + 0.118·11-s − 0.711·13-s + 0.290·14-s + 4.03·16-s − 0.504·17-s − 0.219·19-s + 1.25·20-s + 0.231·22-s − 1.28·23-s + 0.200·25-s − 1.38·26-s + 0.416·28-s + 0.185·29-s − 1.82·31-s + 4.36·32-s − 0.982·34-s + 0.0665·35-s − 1.20·37-s − 0.428·38-s + 1.56·40-s + 0.258·41-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

\(L(1)\) \(\approx\) \(6.037967476\)
\(L(\frac12)\) \(\approx\) \(6.037967476\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.75T + 2T^{2} \)
7 \( 1 - 0.393T + 7T^{2} \)
11 \( 1 - 0.393T + 11T^{2} \)
13 \( 1 + 2.56T + 13T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + 0.958T + 19T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 7.34T + 37T^{2} \)
41 \( 1 - 1.65T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 - 9.54T + 59T^{2} \)
61 \( 1 + 6.25T + 61T^{2} \)
67 \( 1 + 7.42T + 67T^{2} \)
71 \( 1 + 5.98T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 + 6.41T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−9.998729396058834736373674510196, −8.773854017889706828390774493227, −7.48763439957280655097502882799, −7.00019248643794014659916300158, −5.93146675114601501052917610373, −5.48787434677885647395306902480, −4.46522086381216913223976409713, −3.79887005819955356746062876026, −2.59164670677942975261980067788, −1.84436193522081983643835915560, 1.84436193522081983643835915560, 2.59164670677942975261980067788, 3.79887005819955356746062876026, 4.46522086381216913223976409713, 5.48787434677885647395306902480, 5.93146675114601501052917610373, 7.00019248643794014659916300158, 7.48763439957280655097502882799, 8.773854017889706828390774493227, 9.998729396058834736373674510196

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