Properties

Label 2-364650-1.1-c1-0-113
Degree $2$
Conductor $364650$
Sign $-1$
Analytic cond. $2911.74$
Root an. cond. $53.9605$
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-s + 3-s + 4-s − 6-s + 5·7-s − 8-s + 9-s + 11-s + 12-s − 13-s − 5·14-s + 16-s + 17-s − 18-s − 2·19-s + 5·21-s − 22-s − 6·23-s − 24-s + 26-s + 27-s + 5·28-s − 9·29-s − 4·31-s − 32-s + 33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s + 1.09·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.944·28-s − 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(2911.74\)
Root analytic conductor: \(53.9605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 364650,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.69662882689257, −12.23099855565321, −11.58550991976146, −11.36426056827178, −11.02562311418279, −10.36554948613188, −10.06994455925170, −9.496894841309464, −8.904928577891626, −8.735595334746856, −8.118636100235843, −7.809652171142442, −7.379696406332859, −7.091905737946121, −6.248146485736833, −5.728009793272378, −5.349798406176489, −4.644673748570115, −4.219410614944867, −3.747188954918908, −3.089919982724317, −2.279684779699735, −1.879655009435477, −1.642302608572561, −0.8578836176088905, 0, 0.8578836176088905, 1.642302608572561, 1.879655009435477, 2.279684779699735, 3.089919982724317, 3.747188954918908, 4.219410614944867, 4.644673748570115, 5.349798406176489, 5.728009793272378, 6.248146485736833, 7.091905737946121, 7.379696406332859, 7.809652171142442, 8.118636100235843, 8.735595334746856, 8.904928577891626, 9.496894841309464, 10.06994455925170, 10.36554948613188, 11.02562311418279, 11.36426056827178, 11.58550991976146, 12.23099855565321, 12.69662882689257

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