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Properties

Label 2-342-171.110-c1-0-4
Degree $2$
Conductor $342$
Sign $-0.703 - 0.710i$
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 + (1.16 + 1.27i)3-s + (0.173 − 0.984i)4-s + (−0.846 + 2.32i)5-s + (−1.71 − 0.227i)6-s + (1.68 + 2.91i)7-s + (0.500 + 0.866i)8-s + (−0.266 + 2.98i)9-s + (−0.846 − 2.32i)10-s − 1.87i·11-s + (1.46 − 0.929i)12-s + (−2.43 − 6.67i)13-s + (−3.16 − 1.15i)14-s + (−3.96 + 1.63i)15-s + (−0.939 − 0.342i)16-s + (−1.76 + 4.84i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.674 + 0.737i)3-s + (0.0868 − 0.492i)4-s + (−0.378 + 1.04i)5-s + (−0.700 − 0.0928i)6-s + (0.636 + 1.10i)7-s + (0.176 + 0.306i)8-s + (−0.0887 + 0.996i)9-s + (−0.267 − 0.735i)10-s − 0.564i·11-s + (0.421 − 0.268i)12-s + (−0.674 − 1.85i)13-s + (−0.846 − 0.308i)14-s + (−1.02 + 0.422i)15-s + (−0.234 − 0.0855i)16-s + (−0.428 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.703 - 0.710i$
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: \((2,\ 342,\ (\ :1/2),\ -0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451020 + 1.08151i\)
\(L(\frac12)\) \(\approx\) \(0.451020 + 1.08151i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 - 1.27i)T \)
19 \( 1 + (-4.24 - 1.00i)T \)
good5 \( 1 + (0.846 - 2.32i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.68 - 2.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.87iT - 11T^{2} \)
13 \( 1 + (2.43 + 6.67i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.76 - 4.84i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.38 + 0.244i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.0802 + 0.455i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 0.245iT - 31T^{2} \)
37 \( 1 + 2.64iT - 37T^{2} \)
41 \( 1 + (2.57 - 2.15i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.540 - 3.06i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-6.44 - 1.13i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-7.46 - 6.26i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (1.41 + 7.99i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.05 + 3.29i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (3.40 - 4.06i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-10.9 + 9.19i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.08 + 6.18i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.87 + 5.14i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (10.3 - 5.96i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.229 - 1.30i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-11.0 - 13.1i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.50725710277487830583829451956, −10.68391486988627991999720052830, −10.07603690828260315164832652622, −8.920605982480839754425254593715, −8.111728042286400499181644255703, −7.54767064960344731123524263474, −5.96695145173280931162239724586, −5.13529631839526305654083856173, −3.45409516452512621541406197437, −2.43746767833494255238967621868, 0.958132781966907401669934088814, 2.20034740408375763716688486516, 3.98649880821535154742862558427, 4.81077446112784212594913348931, 7.11233997009061979993601544880, 7.26707583067511079909958813500, 8.502838911872026856203225911646, 9.187829378031937323635208034369, 10.00366607945505107029948239379, 11.62702259248899129030137568383

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