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Properties

Label 2-342-171.110-c1-0-13
Degree $2$
Conductor $342$
Sign $0.975 + 0.221i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (1.45 + 0.935i)3-s + (0.173 − 0.984i)4-s + (1.19 − 3.27i)5-s + (−1.71 + 0.220i)6-s + (−0.586 − 1.01i)7-s + (0.500 + 0.866i)8-s + (1.24 + 2.72i)9-s + (1.19 + 3.27i)10-s − 3.78i·11-s + (1.17 − 1.27i)12-s + (−0.288 − 0.791i)13-s + (1.10 + 0.401i)14-s + (4.80 − 3.65i)15-s + (−0.939 − 0.342i)16-s + (0.903 − 2.48i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.841 + 0.540i)3-s + (0.0868 − 0.492i)4-s + (0.533 − 1.46i)5-s + (−0.701 + 0.0898i)6-s + (−0.221 − 0.383i)7-s + (0.176 + 0.306i)8-s + (0.416 + 0.909i)9-s + (0.376 + 1.03i)10-s − 1.14i·11-s + (0.339 − 0.367i)12-s + (−0.0798 − 0.219i)13-s + (0.294 + 0.107i)14-s + (1.24 − 0.944i)15-s + (−0.234 − 0.0855i)16-s + (0.219 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40054 - 0.157344i\)
\(L(\frac12)\) \(\approx\) \(1.40054 - 0.157344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-1.45 - 0.935i)T \)
19 \( 1 + (4.28 + 0.795i)T \)
good5 \( 1 + (-1.19 + 3.27i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.586 + 1.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 + (0.288 + 0.791i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.903 + 2.48i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-8.36 - 1.47i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.629 - 3.57i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 - 8.90iT - 31T^{2} \)
37 \( 1 - 6.66iT - 37T^{2} \)
41 \( 1 + (-4.47 + 3.75i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.88 - 10.6i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.87 + 1.21i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.39 + 1.17i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.670 - 3.80i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.94 + 0.709i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.826 - 0.984i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.811 + 0.680i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.93 + 10.9i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (3.38 - 9.31i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.06 - 0.616i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.846 - 4.80i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (9.42 + 11.2i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.17003735535286875420967574406, −10.28037621913316611604938453980, −9.287844154821212469976676248893, −8.801530580701999891491145680570, −8.119042721372893629433013605946, −6.84097661043036322022320651578, −5.41121589413516920931988390098, −4.68014036536880932800507364727, −3.09787582121922726606430509239, −1.21327905456525186663350565713, 2.04496751649407925575849640214, 2.71202130603197270486143718433, 4.01483677946416654384933580278, 6.14669632115779120592640000342, 6.98044049097502605582179142551, 7.70841622917192573310258157876, 8.950863877872921042007104350310, 9.702607701335006185121370715381, 10.47430547177341437534514181232, 11.41428907204057245055384478826

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