Properties

Label 2-1352-8.3-c0-0-4
Degree $2$
Conductor $1352$
Sign $1$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
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 − 1.80·3-s + 4-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·11-s − 1.80·12-s + 16-s − 0.445·17-s + 2.24·18-s + 1.24·19-s − 0.445·22-s − 1.80·24-s + 25-s − 2.24·27-s + 32-s + 0.801·33-s − 0.445·34-s + 2.24·36-s + 1.24·38-s + 1.24·41-s + 1.24·43-s − 0.445·44-s − 1.80·48-s + 49-s + 50-s + ⋯
L(s)  = 1  + 2-s − 1.80·3-s + 4-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·11-s − 1.80·12-s + 16-s − 0.445·17-s + 2.24·18-s + 1.24·19-s − 0.445·22-s − 1.80·24-s + 25-s − 2.24·27-s + 32-s + 0.801·33-s − 0.445·34-s + 2.24·36-s + 1.24·38-s + 1.24·41-s + 1.24·43-s − 0.445·44-s − 1.80·48-s + 49-s + 50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (339, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218224212\)
\(L(\frac12)\) \(\approx\) \(1.218224212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + 1.80T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 0.445T + T^{2} \)
17 \( 1 + 0.445T + T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.80T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.80T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−10.31622996949810291538832858526, −9.215949014375014229922164153859, −7.62823286732801751367162785335, −7.11971171534175794957419728522, −6.16892165490899244271613806479, −5.63696719239523866602683037939, −4.86621300700599188181954170401, −4.19083423697713425188602266253, −2.81500047103094009942156870388, −1.23776052890527165988518872093, 1.23776052890527165988518872093, 2.81500047103094009942156870388, 4.19083423697713425188602266253, 4.86621300700599188181954170401, 5.63696719239523866602683037939, 6.16892165490899244271613806479, 7.11971171534175794957419728522, 7.62823286732801751367162785335, 9.215949014375014229922164153859, 10.31622996949810291538832858526

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