Properties

Label 2-392-8.3-c0-0-1
Degree $2$
Conductor $392$
Sign $1$
Analytic cond. $0.195633$
Root an. cond. $0.442304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.41·17-s − 1.00·18-s − 1.41·19-s − 1.41·24-s + 25-s − 32-s + 1.41·34-s + 1.00·36-s + 1.41·38-s + 1.41·41-s + 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.00·72-s + ⋯
L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.41·17-s − 1.00·18-s − 1.41·19-s − 1.41·24-s + 25-s − 32-s + 1.41·34-s + 1.00·36-s + 1.41·38-s + 1.41·41-s + 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.00·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.195633\)
Root analytic conductor: \(0.442304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8077912574\)
\(L(\frac12)\) \(\approx\) \(0.8077912574\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−11.15767815449137423742214366434, −10.49256219553940935271304948290, −9.319089283845472391981756214243, −8.836505842614331460046430730780, −8.123806088468366138004013113671, −7.16453561690616187677470284519, −6.21258376867662354644552691581, −4.32698307489476940748726754712, −2.93752529136674474231738964203, −2.00986025874645470728683418307, 2.00986025874645470728683418307, 2.93752529136674474231738964203, 4.32698307489476940748726754712, 6.21258376867662354644552691581, 7.16453561690616187677470284519, 8.123806088468366138004013113671, 8.836505842614331460046430730780, 9.319089283845472391981756214243, 10.49256219553940935271304948290, 11.15767815449137423742214366434

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