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Properties

Label 2-342-171.106-c1-0-5
Degree $2$
Conductor $342$
Sign $-0.983 - 0.182i$
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 + (−0.511 + 1.65i)3-s + (−0.499 + 0.866i)4-s + 0.149·5-s + (−1.68 + 0.384i)6-s + (−0.733 + 1.27i)7-s − 0.999·8-s + (−2.47 − 1.69i)9-s + (0.0748 + 0.129i)10-s + (−1.57 + 2.73i)11-s + (−1.17 − 1.27i)12-s + (−1.16 + 2.02i)13-s − 1.46·14-s + (−0.0764 + 0.247i)15-s + (−0.5 − 0.866i)16-s + (0.345 − 0.598i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.295 + 0.955i)3-s + (−0.249 + 0.433i)4-s + 0.0669·5-s + (−0.689 + 0.157i)6-s + (−0.277 + 0.480i)7-s − 0.353·8-s + (−0.825 − 0.563i)9-s + (0.0236 + 0.0409i)10-s + (−0.476 + 0.824i)11-s + (−0.339 − 0.366i)12-s + (−0.323 + 0.561i)13-s − 0.392·14-s + (−0.0197 + 0.0639i)15-s + (−0.125 − 0.216i)16-s + (0.0837 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.983 - 0.182i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102167 + 1.10783i\)
\(L(\frac12)\) \(\approx\) \(0.102167 + 1.10783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.511 - 1.65i)T \)
19 \( 1 + (-4.31 - 0.637i)T \)
good5 \( 1 - 0.149T + 5T^{2} \)
7 \( 1 + (0.733 - 1.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.57 - 2.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.16 - 2.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.345 + 0.598i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.41 + 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + (-3.37 - 5.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.86T + 37T^{2} \)
41 \( 1 + 0.586T + 41T^{2} \)
43 \( 1 + (-4.32 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.377T + 47T^{2} \)
53 \( 1 + (0.204 + 0.354i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 - 8.43T + 61T^{2} \)
67 \( 1 + (-1.79 + 3.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.861 - 1.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.499 + 0.865i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.24 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.15 - 2.00i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.57 + 2.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.970 + 1.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.98999220915632429610703551873, −11.10571023977054377549200619189, −9.795791590172043335466963972967, −9.424980341697758564883677508564, −8.191037395118449359057650771945, −7.06483771636408041579661221163, −5.92834265033264026369518524042, −5.08026130495588065975115043033, −4.13286575633738427978810875541, −2.76297017889288421590806222920, 0.71880178617805908496666231935, 2.42609031514478352612927004299, 3.64728307803885989113741981060, 5.31307982734110359700214643166, 5.99598977556802072112223815547, 7.31344161285506180075685932781, 8.072968700299484514253453511215, 9.408461563023343372396546015378, 10.39763031912940609568340601824, 11.32950161320373991141122057025

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