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Properties

Label 2-342-171.106-c1-0-2
Degree $2$
Conductor $342$
Sign $0.521 - 0.853i$
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 + (−0.927 − 1.46i)3-s + (−0.499 + 0.866i)4-s − 1.60·5-s + (−0.803 + 1.53i)6-s + (−0.448 + 0.776i)7-s + 0.999·8-s + (−1.28 + 2.71i)9-s + (0.803 + 1.39i)10-s + (−2.43 + 4.22i)11-s + (1.73 − 0.0715i)12-s + (2.36 − 4.09i)13-s + 0.896·14-s + (1.48 + 2.35i)15-s + (−0.5 − 0.866i)16-s + (−3.38 + 5.86i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.535 − 0.844i)3-s + (−0.249 + 0.433i)4-s − 0.718·5-s + (−0.327 + 0.626i)6-s + (−0.169 + 0.293i)7-s + 0.353·8-s + (−0.426 + 0.904i)9-s + (0.254 + 0.439i)10-s + (−0.735 + 1.27i)11-s + (0.499 − 0.0206i)12-s + (0.656 − 1.13i)13-s + 0.239·14-s + (0.384 + 0.606i)15-s + (−0.125 − 0.216i)16-s + (−0.821 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.521 - 0.853i$
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: \((2,\ 342,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323757 + 0.181515i\)
\(L(\frac12)\) \(\approx\) \(0.323757 + 0.181515i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.927 + 1.46i)T \)
19 \( 1 + (-3.41 - 2.70i)T \)
good5 \( 1 + 1.60T + 5T^{2} \)
7 \( 1 + (0.448 - 0.776i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.43 - 4.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.36 + 4.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.38 - 5.86i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.57 + 2.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.461T + 29T^{2} \)
31 \( 1 + (-4.34 - 7.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.97T + 37T^{2} \)
41 \( 1 + 5.02T + 41T^{2} \)
43 \( 1 + (-1.50 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.709T + 47T^{2} \)
53 \( 1 + (-0.134 - 0.233i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + (-4.35 + 7.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.10 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.73 - 9.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.33 + 5.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.541 - 0.937i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.66 + 9.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.93 + 8.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.80727966592430016629419182579, −10.67537275896844347480965138433, −10.27026255352730541899470857185, −8.644702114113208673081467776844, −7.973667045503762091470932720707, −7.11080768050122364411984766591, −5.86400980379662550276014762282, −4.63847230403562449400017286939, −3.12828320122765951884978060761, −1.65415657149132950508237746120, 0.31339177263991485321919601408, 3.30487360396550194019322990322, 4.45974922080347531961211178434, 5.46618675168283565296835111939, 6.55115628822177444793310790200, 7.52534989228121127131222673793, 8.752177938601107100264846822693, 9.337516204516492341723319795470, 10.48452219463672265800927150146, 11.40104794512220605148127701177

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