Properties

Label 2-1305-1.1-c1-0-44
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  + 1.61·2-s + 0.618·4-s + 5-s − 4.23·7-s − 2.23·8-s + 1.61·10-s + 0.236·11-s + 13-s − 6.85·14-s − 4.85·16-s − 7.47·17-s + 2.47·19-s + 0.618·20-s + 0.381·22-s − 4.47·23-s + 25-s + 1.61·26-s − 2.61·28-s + 29-s − 8·31-s − 3.38·32-s − 12.0·34-s − 4.23·35-s + 4.00·38-s − 2.23·40-s + 6·41-s − 6·43-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 0.447·5-s − 1.60·7-s − 0.790·8-s + 0.511·10-s + 0.0711·11-s + 0.277·13-s − 1.83·14-s − 1.21·16-s − 1.81·17-s + 0.567·19-s + 0.138·20-s + 0.0814·22-s − 0.932·23-s + 0.200·25-s + 0.317·26-s − 0.494·28-s + 0.185·29-s − 1.43·31-s − 0.597·32-s − 2.07·34-s − 0.716·35-s + 0.648·38-s − 0.353·40-s + 0.937·41-s − 0.914·43-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 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 - 0.236T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 7.47T + 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 3.76T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 8.47T + 61T^{2} \)
67 \( 1 + 1.29T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 - 9.94T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−9.283736329164674942594386622762, −8.697350228581384984948209950044, −7.22980518118357810536733941944, −6.35972049867194118325046120142, −6.01697766949077985965701528804, −4.95477740032655756884739158408, −3.97290442548443611623593797787, −3.24705627090318953558557404228, −2.23557678949602228280205533743, 0, 2.23557678949602228280205533743, 3.24705627090318953558557404228, 3.97290442548443611623593797787, 4.95477740032655756884739158408, 6.01697766949077985965701528804, 6.35972049867194118325046120142, 7.22980518118357810536733941944, 8.697350228581384984948209950044, 9.283736329164674942594386622762

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