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Properties

Label 2-1352-104.51-c0-0-4
Degree $2$
Conductor $1352$
Sign $0.722 + 0.691i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 0.445·3-s − 4-s − 0.445i·6-s − i·8-s − 0.801·9-s − 1.24i·11-s + 0.445·12-s + 16-s − 1.24·17-s − 0.801i·18-s − 1.80i·19-s + 1.24·22-s + 0.445i·24-s − 25-s + ⋯
L(s)  = 1  + i·2-s − 0.445·3-s − 4-s − 0.445i·6-s − i·8-s − 0.801·9-s − 1.24i·11-s + 0.445·12-s + 16-s − 1.24·17-s − 0.801i·18-s − 1.80i·19-s + 1.24·22-s + 0.445i·24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ 0.722 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4882009113\)
\(L(\frac12)\) \(\approx\) \(0.4882009113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.222272604700825414975218632720, −8.921417313269397070545806453949, −8.100337464027994103928103735069, −7.11953938325727535999094208612, −6.33453463396801328030075653201, −5.69961509932885141558085266650, −4.89842492400367016245373284061, −3.89092477786662805380400267631, −2.67136697427295605255881283989, −0.42908054578575619489261451283, 1.66555128094117912774824049141, 2.64528773367045765537723150750, 3.89914470391984171123558209904, 4.64533563571004823564076980588, 5.61338768662413786392304881539, 6.41696700106877871011564398339, 7.71647743524787743141667132187, 8.381540002770040586458296308086, 9.402475891996781695289867112525, 9.921233919363211990065680118894

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