Properties

Label 2-270802-1.1-c1-0-15
Degree $2$
Conductor $270802$
Sign $-1$
Analytic cond. $2162.36$
Root an. cond. $46.5012$
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 − 2·3-s + 4-s − 2·6-s + 7-s + 8-s + 9-s − 4·11-s − 2·12-s + 14-s + 16-s − 6·17-s + 18-s + 6·19-s − 2·21-s − 4·22-s − 23-s − 2·24-s − 5·25-s + 4·27-s + 28-s − 4·31-s + 32-s + 8·33-s − 6·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.37·19-s − 0.436·21-s − 0.852·22-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270802\)    =    \(2 \cdot 7 \cdot 23 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2162.36\)
Root analytic conductor: \(46.5012\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{270802} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.08861620550862, −12.56475838308165, −12.02075217445179, −11.52262441191434, −11.34400666076151, −10.81654923446204, −10.57627104319503, −9.891754299471884, −9.420481278996715, −8.893759672954321, −8.104223827302884, −7.782388317743946, −7.339775550084347, −6.738124134170632, −6.218876526748676, −5.781688411234480, −5.423808394771537, −4.872155699137261, −4.592854131977405, −3.971633389672247, −3.260213249586595, −2.700255401905206, −2.143376759615269, −1.508177191702510, −0.6505723135462482, 0, 0.6505723135462482, 1.508177191702510, 2.143376759615269, 2.700255401905206, 3.260213249586595, 3.971633389672247, 4.592854131977405, 4.872155699137261, 5.423808394771537, 5.781688411234480, 6.218876526748676, 6.738124134170632, 7.339775550084347, 7.782388317743946, 8.104223827302884, 8.893759672954321, 9.420481278996715, 9.891754299471884, 10.57627104319503, 10.81654923446204, 11.34400666076151, 11.52262441191434, 12.02075217445179, 12.56475838308165, 13.08861620550862

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