Properties

Label 2-624e2-1.1-c1-0-4
Degree $2$
Conductor $389376$
Sign $1$
Analytic cond. $3109.18$
Root an. cond. $55.7600$
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  − 2·5-s + 2·17-s − 25-s − 4·29-s + 2·37-s − 8·41-s − 7·49-s + 4·53-s + 12·61-s − 16·73-s − 4·85-s + 16·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.485·17-s − 1/5·25-s − 0.742·29-s + 0.328·37-s − 1.24·41-s − 49-s + 0.549·53-s + 1.53·61-s − 1.87·73-s − 0.433·85-s + 1.69·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(389376\)    =    \(2^{8} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3109.18\)
Root analytic conductor: \(55.7600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 389376,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7437493018\)
\(L(\frac12)\) \(\approx\) \(0.7437493018\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.25546127955879, −11.97289093990341, −11.66177227028994, −11.12539355375774, −10.78440326746831, −10.09488518580366, −9.875127868511230, −9.271224357569998, −8.767635488781683, −8.289941253445130, −7.918104594170816, −7.449403840639517, −7.053821085287925, −6.494306245657989, −5.999356486223569, −5.380234401449326, −5.035498087663906, −4.385413539977461, −3.862728432720932, −3.548411277222912, −2.950669994195244, −2.335866259353213, −1.660425603093515, −1.070416257969322, −0.2386474050756605, 0.2386474050756605, 1.070416257969322, 1.660425603093515, 2.335866259353213, 2.950669994195244, 3.548411277222912, 3.862728432720932, 4.385413539977461, 5.035498087663906, 5.380234401449326, 5.999356486223569, 6.494306245657989, 7.053821085287925, 7.449403840639517, 7.918104594170816, 8.289941253445130, 8.767635488781683, 9.271224357569998, 9.875127868511230, 10.09488518580366, 10.78440326746831, 11.12539355375774, 11.66177227028994, 11.97289093990341, 12.25546127955879

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