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Properties

Label 2-342-171.65-c1-0-8
Degree $2$
Conductor $342$
Sign $-0.426 - 0.904i$
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  + (0.5 + 0.866i)2-s + (1.27 + 1.16i)3-s + (−0.499 + 0.866i)4-s − 0.317i·5-s + (−0.370 + 1.69i)6-s + (−1.37 + 2.37i)7-s − 0.999·8-s + (0.275 + 2.98i)9-s + (0.275 − 0.158i)10-s + (0.427 + 0.247i)11-s + (−1.65 + 0.524i)12-s + (−3.13 − 1.80i)13-s − 2.74·14-s + (0.370 − 0.406i)15-s + (−0.5 − 0.866i)16-s + (4.85 + 2.80i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.738 + 0.673i)3-s + (−0.249 + 0.433i)4-s − 0.142i·5-s + (−0.151 + 0.690i)6-s + (−0.518 + 0.897i)7-s − 0.353·8-s + (0.0917 + 0.995i)9-s + (0.0870 − 0.0502i)10-s + (0.128 + 0.0744i)11-s + (−0.476 + 0.151i)12-s + (−0.869 − 0.501i)13-s − 0.732·14-s + (0.0957 − 0.105i)15-s + (−0.125 − 0.216i)16-s + (1.17 + 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970203 + 1.52935i\)
\(L(\frac12)\) \(\approx\) \(0.970203 + 1.52935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.27 - 1.16i)T \)
19 \( 1 + (-2.54 + 3.54i)T \)
good5 \( 1 + 0.317iT - 5T^{2} \)
7 \( 1 + (1.37 - 2.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.427 - 0.247i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.13 + 1.80i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.85 - 2.80i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.72 - 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.71T + 29T^{2} \)
31 \( 1 + (-7.64 + 4.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + 3.85T + 41T^{2} \)
43 \( 1 + (2.65 + 4.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.90iT - 47T^{2} \)
53 \( 1 + (-2.35 - 4.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 + (-2.86 - 1.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.72 + 11.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.11 + 5.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.273 + 0.158i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.66 + 0.960i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.39 - 2.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.17 - 2.41i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.06544544189165148467839502522, −10.77494492654987784787400802780, −9.578809397425542255065589872479, −9.144858579645294714004041382896, −8.036085516874234212031076096248, −7.20207432405954300229984539398, −5.70901255347547886140279307623, −5.01008091009319742099257749292, −3.62215588096295043869517992419, −2.62283189183717878800942005241, 1.18209414828668863777618959408, 2.84253406704764363135293938793, 3.67622639253191639419350981308, 5.08948033279577648168696471139, 6.63839183034394598661064425789, 7.28883512883065820720264859533, 8.428550940135444508300889908935, 9.690325720918054759677613311596, 10.06607254488864058579779663381, 11.47707533209593649771781870009

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