Properties

Label 2-1305-1.1-c3-0-98
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 + 4·7-s − 15·8-s − 5·10-s + 36·11-s − 22·13-s + 4·14-s + 41·16-s + 2·17-s − 56·19-s + 35·20-s + 36·22-s + 40·23-s + 25·25-s − 22·26-s − 28·28-s − 29·29-s + 152·31-s + 161·32-s + 2·34-s − 20·35-s + 34·37-s − 56·38-s + 75·40-s + 250·41-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 0.447·5-s + 0.215·7-s − 0.662·8-s − 0.158·10-s + 0.986·11-s − 0.469·13-s + 0.0763·14-s + 0.640·16-s + 0.0285·17-s − 0.676·19-s + 0.391·20-s + 0.348·22-s + 0.362·23-s + 1/5·25-s − 0.165·26-s − 0.188·28-s − 0.185·29-s + 0.880·31-s + 0.889·32-s + 0.0100·34-s − 0.0965·35-s + 0.151·37-s − 0.239·38-s + 0.296·40-s + 0.952·41-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 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 + 22 T + p^{3} T^{2} \)
17 \( 1 - 2 T + p^{3} T^{2} \)
19 \( 1 + 56 T + p^{3} T^{2} \)
23 \( 1 - 40 T + p^{3} T^{2} \)
31 \( 1 - 152 T + p^{3} T^{2} \)
37 \( 1 - 34 T + p^{3} T^{2} \)
41 \( 1 - 250 T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 - 120 T + p^{3} T^{2} \)
53 \( 1 - 762 T + p^{3} T^{2} \)
59 \( 1 - 188 T + p^{3} T^{2} \)
61 \( 1 + 54 T + p^{3} T^{2} \)
67 \( 1 + 244 T + p^{3} T^{2} \)
71 \( 1 + 600 T + p^{3} T^{2} \)
73 \( 1 - 6 T + p^{3} T^{2} \)
79 \( 1 + 640 T + p^{3} T^{2} \)
83 \( 1 + 8 p T + p^{3} T^{2} \)
89 \( 1 + 150 T + p^{3} T^{2} \)
97 \( 1 + 1690 T + p^{3} T^{2} \)
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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

−8.799274092144484034010022378415, −8.243784509853128711624229776506, −7.20713917104066498006599468055, −6.32182511315958602501595330440, −5.34323381248676301603905629226, −4.44490478136411093815197721043, −3.88882117486361738448878173972, −2.76753115940428069465032111367, −1.22755963942782866110149577146, 0, 1.22755963942782866110149577146, 2.76753115940428069465032111367, 3.88882117486361738448878173972, 4.44490478136411093815197721043, 5.34323381248676301603905629226, 6.32182511315958602501595330440, 7.20713917104066498006599468055, 8.243784509853128711624229776506, 8.799274092144484034010022378415

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