Properties

Label 2-585-1.1-c1-0-15
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + 2.56·2-s + 4.56·4-s + 5-s − 0.438·7-s + 6.56·8-s + 2.56·10-s − 1.56·11-s − 13-s − 1.12·14-s + 7.68·16-s + 1.56·17-s − 5.12·19-s + 4.56·20-s − 4·22-s + 2.43·23-s + 25-s − 2.56·26-s − 2·28-s − 7.12·29-s + 6·31-s + 6.56·32-s + 4·34-s − 0.438·35-s − 10.6·37-s − 13.1·38-s + 6.56·40-s + 3.56·41-s + ⋯
L(s)  = 1  + 1.81·2-s + 2.28·4-s + 0.447·5-s − 0.165·7-s + 2.31·8-s + 0.810·10-s − 0.470·11-s − 0.277·13-s − 0.300·14-s + 1.92·16-s + 0.378·17-s − 1.17·19-s + 1.01·20-s − 0.852·22-s + 0.508·23-s + 0.200·25-s − 0.502·26-s − 0.377·28-s − 1.32·29-s + 1.07·31-s + 1.15·32-s + 0.685·34-s − 0.0741·35-s − 1.75·37-s − 2.12·38-s + 1.03·40-s + 0.556·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.194333390\)
\(L(\frac12)\) \(\approx\) \(4.194333390\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 - 2.56T + 2T^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 - 3.12T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 4.68T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 6.68T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 4.68T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−10.88646095615845168123369960683, −10.23759678723276250291360854061, −8.960271906157408174203070085830, −7.63962013293101031886046918865, −6.74099003219979492542022846742, −5.88757229265970843563261417573, −5.12356601790988876342595548419, −4.16458282532561182364840712696, −3.05987472392565070404397409404, −2.03545650757098222040794671648, 2.03545650757098222040794671648, 3.05987472392565070404397409404, 4.16458282532561182364840712696, 5.12356601790988876342595548419, 5.88757229265970843563261417573, 6.74099003219979492542022846742, 7.63962013293101031886046918865, 8.960271906157408174203070085830, 10.23759678723276250291360854061, 10.88646095615845168123369960683

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