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Properties

Label 2-1352-104.35-c0-0-3
Degree $2$
Conductor $1352$
Sign $0.488 - 0.872i$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)10-s + 0.999·12-s − 0.999·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.499i)20-s + 0.999i·21-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)10-s + 0.999·12-s − 0.999·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.499i)20-s + 0.999i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ 0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.902426545\)
\(L(\frac12)\) \(\approx\) \(1.902426545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.878223563661038460325055686557, −8.924852405632046828378905518650, −7.971085016617006416533514411200, −7.30157786417098438273091637500, −6.65520050365649572652023475850, −6.10944561910638960081987133091, −4.98053795137731331921032491752, −3.75559546066981199487332866595, −2.75626672508052850615022171515, −2.29428874240850158571732940594, 1.29676769859631177276471469427, 2.87827314733881036786411533067, 3.71196760942098020342955847462, 4.39561646553212816775993118578, 5.10879997575155783910310188429, 6.23423146601742820600109286327, 6.94955430246636260664819329260, 8.321905772282808814128531332363, 9.140190807241498300615741095123, 9.718282037282881841020902756979

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