Properties

Label 2-1305-1.1-c3-0-119
Degree $2$
Conductor $1305$
Sign $-1$
Analytic cond. $76.9974$
Root an. cond. $8.77482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·4-s − 5·5-s + 29·7-s − 24·8-s − 10·10-s + 15·11-s + 3·13-s + 58·14-s − 16·16-s − 121·17-s − 40·19-s + 20·20-s + 30·22-s + 116·23-s + 25·25-s + 6·26-s − 116·28-s − 29·29-s − 116·31-s + 160·32-s − 242·34-s − 145·35-s + 36·37-s − 80·38-s + 120·40-s + 170·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.56·7-s − 1.06·8-s − 0.316·10-s + 0.411·11-s + 0.0640·13-s + 1.10·14-s − 1/4·16-s − 1.72·17-s − 0.482·19-s + 0.223·20-s + 0.290·22-s + 1.05·23-s + 1/5·25-s + 0.0452·26-s − 0.782·28-s − 0.185·29-s − 0.672·31-s + 0.883·32-s − 1.22·34-s − 0.700·35-s + 0.159·37-s − 0.341·38-s + 0.474·40-s + 0.647·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(76.9974\)
Root analytic conductor: \(8.77482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1305,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
29 \( 1 + p T \)
good2 \( 1 - p T + p^{3} T^{2} \)
7 \( 1 - 29 T + p^{3} T^{2} \)
11 \( 1 - 15 T + p^{3} T^{2} \)
13 \( 1 - 3 T + p^{3} T^{2} \)
17 \( 1 + 121 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 - 116 T + p^{3} T^{2} \)
31 \( 1 + 116 T + p^{3} T^{2} \)
37 \( 1 - 36 T + p^{3} T^{2} \)
41 \( 1 - 170 T + p^{3} T^{2} \)
43 \( 1 - 230 T + p^{3} T^{2} \)
47 \( 1 + 231 T + p^{3} T^{2} \)
53 \( 1 + 456 T + p^{3} T^{2} \)
59 \( 1 + 576 T + p^{3} T^{2} \)
61 \( 1 - 342 T + p^{3} T^{2} \)
67 \( 1 + 269 T + p^{3} T^{2} \)
71 \( 1 + 302 T + p^{3} T^{2} \)
73 \( 1 + 372 T + p^{3} T^{2} \)
79 \( 1 + 348 T + p^{3} T^{2} \)
83 \( 1 - 512 T + p^{3} T^{2} \)
89 \( 1 + 1525 T + p^{3} T^{2} \)
97 \( 1 + 560 T + p^{3} T^{2} \)
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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

−8.800299654940589186754844743845, −8.187181697190161719874406635204, −7.21465625397190763825629740764, −6.24075977875754200617352628989, −5.19334547672254233279361263825, −4.52003501726931051022986062041, −4.01642610006085264676605416727, −2.69667924015604615592450470960, −1.44085671932689272032338509479, 0, 1.44085671932689272032338509479, 2.69667924015604615592450470960, 4.01642610006085264676605416727, 4.52003501726931051022986062041, 5.19334547672254233279361263825, 6.24075977875754200617352628989, 7.21465625397190763825629740764, 8.187181697190161719874406635204, 8.800299654940589186754844743845

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