Properties

Label 2-342-19.17-c1-0-1
Degree $2$
Conductor $342$
Sign $0.363 - 0.931i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−3.20 + 1.16i)5-s + (1.43 + 2.49i)7-s + (−0.5 + 0.866i)8-s + (0.592 + 3.35i)10-s + (−0.173 + 0.300i)11-s + (−1.26 + 1.06i)13-s + (2.70 − 0.984i)14-s + (0.766 + 0.642i)16-s + (−1.20 + 6.83i)17-s + (−2.82 + 3.31i)19-s + 3.41·20-s + (0.266 + 0.223i)22-s + (6.39 + 2.32i)23-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.469 − 0.171i)4-s + (−1.43 + 0.521i)5-s + (0.544 + 0.942i)7-s + (−0.176 + 0.306i)8-s + (0.187 + 1.06i)10-s + (−0.0523 + 0.0906i)11-s + (−0.351 + 0.294i)13-s + (0.723 − 0.263i)14-s + (0.191 + 0.160i)16-s + (−0.292 + 1.65i)17-s + (−0.648 + 0.761i)19-s + 0.762·20-s + (0.0567 + 0.0475i)22-s + (1.33 + 0.484i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.363 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649276 + 0.443509i\)
\(L(\frac12)\) \(\approx\) \(0.649276 + 0.443509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 \)
19 \( 1 + (2.82 - 3.31i)T \)
good5 \( 1 + (3.20 - 1.16i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.26 - 1.06i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.20 - 6.83i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-6.39 - 2.32i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.10 + 6.25i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.798 + 1.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + (-2.67 - 2.24i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.14 - 0.780i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.971 + 5.51i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.86 - 0.677i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.0773 - 0.438i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-11.7 - 4.28i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.187 + 1.06i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (15.6 - 5.68i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-9.51 - 7.98i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (8.36 + 7.02i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-5.85 - 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.37 - 1.15i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.634 + 3.59i)T + (-91.1 - 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66731640984066588711693958591, −11.02714407944396087691065677110, −10.12613039504899124993312819500, −8.720892352183291175164616301909, −8.197795492086977213219976355308, −7.03392020792913523143949331711, −5.66839738357660160076818921451, −4.37625241581065770391725423144, −3.48859390962671326085007145228, −2.04845337627038286289685327864, 0.53585820365205604998633771270, 3.30662777913372323610006172515, 4.58205892393542977250861348007, 5.02754425349794018141874607740, 7.06889750400559718000430660634, 7.27755476003830280828201861463, 8.425611095465785794262616833535, 9.102938483021773718450319585027, 10.62816445644600931927662752576, 11.35087801982761038487791737971

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