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Properties

Label 2-342-171.86-c1-0-6
Degree $2$
Conductor $342$
Sign $0.904 - 0.427i$
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 + (−0.210 + 1.71i)3-s + (0.173 − 0.984i)4-s + (0.649 + 0.774i)5-s + (0.944 + 1.45i)6-s + 2.88·7-s + (−0.500 − 0.866i)8-s + (−2.91 − 0.722i)9-s + (0.995 + 0.175i)10-s + (0.354 + 0.204i)11-s + (1.65 + 0.505i)12-s + (1.10 + 3.03i)13-s + (2.21 − 1.85i)14-s + (−1.46 + 0.954i)15-s + (−0.939 − 0.342i)16-s + (1.64 − 0.289i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.121 + 0.992i)3-s + (0.0868 − 0.492i)4-s + (0.290 + 0.346i)5-s + (0.385 + 0.592i)6-s + 1.09·7-s + (−0.176 − 0.306i)8-s + (−0.970 − 0.240i)9-s + (0.314 + 0.0555i)10-s + (0.106 + 0.0617i)11-s + (0.478 + 0.145i)12-s + (0.306 + 0.842i)13-s + (0.591 − 0.496i)14-s + (−0.378 + 0.246i)15-s + (−0.234 − 0.0855i)16-s + (0.398 − 0.0702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.904 - 0.427i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84860 + 0.414639i\)
\(L(\frac12)\) \(\approx\) \(1.84860 + 0.414639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.210 - 1.71i)T \)
19 \( 1 + (-0.598 - 4.31i)T \)
good5 \( 1 + (-0.649 - 0.774i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 - 2.88T + 7T^{2} \)
11 \( 1 + (-0.354 - 0.204i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.10 - 3.03i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.64 + 0.289i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.732 + 0.129i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.00 - 1.82i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.11 + 2.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.635iT - 37T^{2} \)
41 \( 1 + (0.485 + 2.75i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.859 + 4.87i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.56 + 12.5i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (11.6 - 4.23i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (6.37 + 2.32i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.61 - 3.03i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.57 + 4.25i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.24 - 0.817i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (6.53 - 5.48i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.10 + 11.2i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + (-5.14 - 4.31i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-10.7 - 12.7i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.56236655579828117449302136155, −10.71074686112151266900094955779, −10.02184544773788435608219990249, −9.030980531410900617547084732579, −7.960757255672979314117294525445, −6.43000695051029470018945867403, −5.42583090116153761435954042025, −4.48701646431506740293093836261, −3.51196065016543280826143413704, −1.94725485361999248313068896047, 1.43140544099715334511741049158, 3.01287265256720602946657111491, 4.78112600116030648011483755157, 5.57110354678380383363970931436, 6.56675594446849108394853064764, 7.74279248300461151942070883256, 8.201120948909822937458155482937, 9.366159035441544748871604769615, 11.00493627314320727206272172562, 11.51856615224317818678104762405

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