Properties

Label 2-1305-1.1-c1-0-45
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 − 3·7-s − 2.23·8-s + 1.61·10-s − 3.47·11-s − 1.76·13-s − 4.85·14-s − 4.85·16-s + 5.47·17-s − 5.70·19-s + 0.618·20-s − 5.61·22-s + 25-s − 2.85·26-s − 1.85·28-s − 29-s − 8·31-s − 3.38·32-s + 8.85·34-s − 3·35-s − 8·37-s − 9.23·38-s − 2.23·40-s − 4.47·41-s − 1.23·43-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 0.447·5-s − 1.13·7-s − 0.790·8-s + 0.511·10-s − 1.04·11-s − 0.489·13-s − 1.29·14-s − 1.21·16-s + 1.32·17-s − 1.30·19-s + 0.138·20-s − 1.19·22-s + 0.200·25-s − 0.559·26-s − 0.350·28-s − 0.185·29-s − 1.43·31-s − 0.597·32-s + 1.51·34-s − 0.507·35-s − 1.31·37-s − 1.49·38-s − 0.353·40-s − 0.698·41-s − 0.188·43-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 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 3.47T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 + 5.70T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 - 6.70T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 0.763T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 + 2.52T + 67T^{2} \)
71 \( 1 + 2.76T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 9.70T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 7.23T + 97T^{2} \)
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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

−9.328163951011667657685019483990, −8.517157331709636875718521010631, −7.33805759860330208546512951483, −6.52717907393875981455482053830, −5.61319524016706448913866919633, −5.20504930187934825297667122058, −3.96029884260651070620463162534, −3.18858244896882346992177232948, −2.26582325386719369073341016964, 0, 2.26582325386719369073341016964, 3.18858244896882346992177232948, 3.96029884260651070620463162534, 5.20504930187934825297667122058, 5.61319524016706448913866919633, 6.52717907393875981455482053830, 7.33805759860330208546512951483, 8.517157331709636875718521010631, 9.328163951011667657685019483990

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