Properties

Label 2-1305-1.1-c1-0-28
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.311·2-s − 1.90·4-s − 5-s − 0.903·7-s + 1.21·8-s + 0.311·10-s + 1.52·11-s − 0.622·13-s + 0.280·14-s + 3.42·16-s + 7.95·17-s − 1.09·19-s + 1.90·20-s − 0.474·22-s − 7.52·23-s + 25-s + 0.193·26-s + 1.71·28-s + 29-s − 6.90·31-s − 3.49·32-s − 2.47·34-s + 0.903·35-s + 3.95·37-s + 0.341·38-s − 1.21·40-s − 3.67·41-s + ⋯
L(s)  = 1  − 0.219·2-s − 0.951·4-s − 0.447·5-s − 0.341·7-s + 0.429·8-s + 0.0983·10-s + 0.459·11-s − 0.172·13-s + 0.0750·14-s + 0.857·16-s + 1.92·17-s − 0.251·19-s + 0.425·20-s − 0.101·22-s − 1.56·23-s + 0.200·25-s + 0.0379·26-s + 0.324·28-s + 0.185·29-s − 1.23·31-s − 0.617·32-s − 0.424·34-s + 0.152·35-s + 0.650·37-s + 0.0553·38-s − 0.192·40-s − 0.573·41-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: \(1\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.311T + 2T^{2} \)
7 \( 1 + 0.903T + 7T^{2} \)
11 \( 1 - 1.52T + 11T^{2} \)
13 \( 1 + 0.622T + 13T^{2} \)
17 \( 1 - 7.95T + 17T^{2} \)
19 \( 1 + 1.09T + 19T^{2} \)
23 \( 1 + 7.52T + 23T^{2} \)
31 \( 1 + 6.90T + 31T^{2} \)
37 \( 1 - 3.95T + 37T^{2} \)
41 \( 1 + 3.67T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 6.90T + 47T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 9.13T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 + 4.08T + 97T^{2} \)
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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

−9.485766913754565268795126136905, −8.167293485534335815797443050122, −8.027986117108595340978255802960, −6.84901370512058031546813207168, −5.80385595673914163595518467365, −4.97634793852087336703707632075, −3.91819856904807039536980891951, −3.30021799707193155491695692432, −1.50019452928248440997371526910, 0, 1.50019452928248440997371526910, 3.30021799707193155491695692432, 3.91819856904807039536980891951, 4.97634793852087336703707632075, 5.80385595673914163595518467365, 6.84901370512058031546813207168, 8.027986117108595340978255802960, 8.167293485534335815797443050122, 9.485766913754565268795126136905

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