Properties

Label 2-1305-1.1-c1-0-36
Degree $2$
Conductor $1305$
Sign $-1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s − 2·7-s − 11-s + 6·13-s + 4·16-s − 4·17-s − 2·19-s − 2·20-s − 3·23-s + 25-s + 4·28-s − 29-s − 4·31-s − 2·35-s − 3·37-s − 7·41-s + 5·43-s + 2·44-s − 6·47-s − 3·49-s − 12·52-s − 13·53-s − 55-s − 8·64-s + 6·65-s − 10·67-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.755·7-s − 0.301·11-s + 1.66·13-s + 16-s − 0.970·17-s − 0.458·19-s − 0.447·20-s − 0.625·23-s + 1/5·25-s + 0.755·28-s − 0.185·29-s − 0.718·31-s − 0.338·35-s − 0.493·37-s − 1.09·41-s + 0.762·43-s + 0.301·44-s − 0.875·47-s − 3/7·49-s − 1.66·52-s − 1.78·53-s − 0.134·55-s − 64-s + 0.744·65-s − 1.22·67-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: yes
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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−9.091450876069161433530704059105, −8.702233568157157082072920489522, −7.78822795874009599925729941525, −6.49354047175213471556403448313, −6.00725269273677695580992834090, −4.97538272189352451761284735335, −3.98895410887896873098017235002, −3.20083113444945361890680615236, −1.65791202301625672384757641334, 0, 1.65791202301625672384757641334, 3.20083113444945361890680615236, 3.98895410887896873098017235002, 4.97538272189352451761284735335, 6.00725269273677695580992834090, 6.49354047175213471556403448313, 7.78822795874009599925729941525, 8.702233568157157082072920489522, 9.091450876069161433530704059105

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