Properties

Label 2-1305-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s + 2·7-s − 3·11-s + 2·13-s + 4·16-s + 2·19-s − 2·20-s − 3·23-s + 25-s − 4·28-s + 29-s + 8·31-s + 2·35-s − 37-s + 3·41-s − 43-s + 6·44-s + 6·47-s − 3·49-s − 4·52-s + 3·53-s − 3·55-s + 12·59-s + 8·61-s − 8·64-s + 2·65-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.755·7-s − 0.904·11-s + 0.554·13-s + 16-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 + 1.43·31-s + 0.338·35-s − 0.164·37-s + 0.468·41-s − 0.152·43-s + 0.904·44-s + 0.875·47-s − 3/7·49-s − 0.554·52-s + 0.412·53-s − 0.404·55-s + 1.56·59-s + 1.02·61-s − 64-s + 0.248·65-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: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.500117804\)
\(L(\frac12)\) \(\approx\) \(1.500117804\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−9.788906746241101084288007436947, −8.642814857707797957392115254585, −8.272248506160292840175320854368, −7.38238509669578907211253317909, −6.12295429159013140281371744854, −5.32855597269579923033160745483, −4.66262168952595729437838996730, −3.65107489984939242445264232066, −2.36748279843542963780814696470, −0.942520552818180520165204014558, 0.942520552818180520165204014558, 2.36748279843542963780814696470, 3.65107489984939242445264232066, 4.66262168952595729437838996730, 5.32855597269579923033160745483, 6.12295429159013140281371744854, 7.38238509669578907211253317909, 8.272248506160292840175320854368, 8.642814857707797957392115254585, 9.788906746241101084288007436947

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