Properties

Label 2-1305-1.1-c3-0-80
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-s − 7·4-s + 5·5-s − 14·7-s + 15·8-s − 5·10-s − 62·11-s + 42·13-s + 14·14-s + 41·16-s + 114·17-s − 70·19-s − 35·20-s + 62·22-s − 62·23-s + 25·25-s − 42·26-s + 98·28-s + 29·29-s + 142·31-s − 161·32-s − 114·34-s − 70·35-s + 146·37-s + 70·38-s + 75·40-s − 162·41-s + ⋯
L(s)  = 1  − 0.353·2-s − 7/8·4-s + 0.447·5-s − 0.755·7-s + 0.662·8-s − 0.158·10-s − 1.69·11-s + 0.896·13-s + 0.267·14-s + 0.640·16-s + 1.62·17-s − 0.845·19-s − 0.391·20-s + 0.600·22-s − 0.562·23-s + 1/5·25-s − 0.316·26-s + 0.661·28-s + 0.185·29-s + 0.822·31-s − 0.889·32-s − 0.575·34-s − 0.338·35-s + 0.648·37-s + 0.298·38-s + 0.296·40-s − 0.617·41-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 \( 1 + T + p^{3} T^{2} \)
7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 + 62 T + p^{3} T^{2} \)
13 \( 1 - 42 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 + 70 T + p^{3} T^{2} \)
23 \( 1 + 62 T + p^{3} T^{2} \)
31 \( 1 - 142 T + p^{3} T^{2} \)
37 \( 1 - 146 T + p^{3} T^{2} \)
41 \( 1 + 162 T + p^{3} T^{2} \)
43 \( 1 - 352 T + p^{3} T^{2} \)
47 \( 1 - 444 T + p^{3} T^{2} \)
53 \( 1 - 238 T + p^{3} T^{2} \)
59 \( 1 + 840 T + p^{3} T^{2} \)
61 \( 1 - 2 T + p^{3} T^{2} \)
67 \( 1 + 154 T + p^{3} T^{2} \)
71 \( 1 + 892 T + p^{3} T^{2} \)
73 \( 1 + 38 T + p^{3} T^{2} \)
79 \( 1 - 1050 T + p^{3} T^{2} \)
83 \( 1 - 778 T + p^{3} T^{2} \)
89 \( 1 + 1410 T + p^{3} T^{2} \)
97 \( 1 - 466 T + p^{3} T^{2} \)
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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

−8.903759663263932520569423817772, −8.092421527382150656695801461593, −7.55025901922051914898240012090, −6.15826053054107176042349925727, −5.61615576589155939218908853167, −4.64071254243350114214711399857, −3.57113140498816848776609100015, −2.59533952827281246141774684318, −1.11212946454411523521506132160, 0, 1.11212946454411523521506132160, 2.59533952827281246141774684318, 3.57113140498816848776609100015, 4.64071254243350114214711399857, 5.61615576589155939218908853167, 6.15826053054107176042349925727, 7.55025901922051914898240012090, 8.092421527382150656695801461593, 8.903759663263932520569423817772

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