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Properties

Label 2-58e2-116.91-c0-0-3
Degree $2$
Conductor $3364$
Sign $0.133 - 0.991i$
Analytic cond. $1.67885$
Root an. cond. $1.29570$
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  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.0990 + 0.433i)5-s + (0.781 + 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.193 + 0.400i)10-s + (−0.777 − 0.974i)13-s + (0.623 + 0.781i)16-s + 1.80i·17-s + (−0.781 + 0.623i)18-s + (−0.277 + 0.347i)20-s + (0.722 + 0.347i)25-s + (−0.541 − 1.12i)26-s + (0.433 + 0.900i)32-s + (−0.400 + 1.75i)34-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.0990 + 0.433i)5-s + (0.781 + 0.623i)8-s + (−0.623 + 0.781i)9-s + (−0.193 + 0.400i)10-s + (−0.777 − 0.974i)13-s + (0.623 + 0.781i)16-s + 1.80i·17-s + (−0.781 + 0.623i)18-s + (−0.277 + 0.347i)20-s + (0.722 + 0.347i)25-s + (−0.541 − 1.12i)26-s + (0.433 + 0.900i)32-s + (−0.400 + 1.75i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.133 - 0.991i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (2759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.133 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.144764277\)
\(L(\frac12)\) \(\approx\) \(2.144764277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.974 - 0.222i)T \)
29 \( 1 \)
good3 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 - 1.80iT - T^{2} \)
19 \( 1 + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (-0.974 - 0.777i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 - 1.24iT - T^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.781 + 1.62i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.541 + 1.12i)T + (-0.623 - 0.781i)T^{2} \)
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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

−8.580399082534289244317947269768, −8.075338709015564681522869616941, −7.45931298093959683836241415871, −6.56471721013032013438347757671, −5.89306109311719448442916029019, −5.18360738886211301481829845153, −4.45575109254573018192528723539, −3.39430841597594618724994506656, −2.78320844030862101533426451034, −1.79988945759659153020290961164, 0.921619506119808891771493647787, 2.39414240511590565518944786430, 2.97927901538524253698747719325, 4.14403765839799000005189194559, 4.66900502895203564277564697369, 5.51558852059715132469644764841, 6.21078180617318162501399101908, 7.14874714258104068729174419685, 7.49352012013356704363198812184, 8.963344460338036367391086477858

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