Properties

Label 2-94640-1.1-c1-0-45
Degree $2$
Conductor $94640$
Sign $1$
Analytic cond. $755.704$
Root an. cond. $27.4900$
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  − 3-s + 5-s + 7-s − 2·9-s − 3·11-s − 15-s + 3·17-s + 2·19-s − 21-s + 6·23-s + 25-s + 5·27-s + 3·29-s − 4·31-s + 3·33-s + 35-s − 2·37-s + 12·41-s + 10·43-s − 2·45-s + 9·47-s + 49-s − 3·51-s + 12·53-s − 3·55-s − 2·57-s + 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.258·15-s + 0.727·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.522·33-s + 0.169·35-s − 0.328·37-s + 1.87·41-s + 1.52·43-s − 0.298·45-s + 1.31·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.404·55-s − 0.264·57-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94640\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(755.704\)
Root analytic conductor: \(27.4900\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{94640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 94640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.752061608\)
\(L(\frac12)\) \(\approx\) \(2.752061608\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.84543103722732, −13.30602194125717, −12.80467091457648, −12.27318328042705, −11.93253075665390, −11.23161059562227, −10.83366149175546, −10.53997043568164, −9.974421642277905, −9.193038899644333, −8.968227615094615, −8.328664729956486, −7.619515395185735, −7.366536526131762, −6.690944761461390, −5.861528702209496, −5.675482062696673, −5.198172135358081, −4.667380854342532, −3.927768316936222, −3.140843944571217, −2.609111624373697, −2.111850810614510, −0.9360205720062219, −0.7083000791511709, 0.7083000791511709, 0.9360205720062219, 2.111850810614510, 2.609111624373697, 3.140843944571217, 3.927768316936222, 4.667380854342532, 5.198172135358081, 5.675482062696673, 5.861528702209496, 6.690944761461390, 7.366536526131762, 7.619515395185735, 8.328664729956486, 8.968227615094615, 9.193038899644333, 9.974421642277905, 10.53997043568164, 10.83366149175546, 11.23161059562227, 11.93253075665390, 12.27318328042705, 12.80467091457648, 13.30602194125717, 13.84543103722732

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