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Properties

Label 2-342-171.121-c1-0-1
Degree $2$
Conductor $342$
Sign $0.323 - 0.946i$
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 + (0.108 + 1.72i)3-s + (−0.499 − 0.866i)4-s − 2.39·5-s + (1.55 + 0.770i)6-s + (0.959 + 1.66i)7-s − 0.999·8-s + (−2.97 + 0.374i)9-s + (−1.19 + 2.07i)10-s + (2.87 + 4.98i)11-s + (1.44 − 0.958i)12-s + (2.30 + 3.98i)13-s + 1.91·14-s + (−0.259 − 4.14i)15-s + (−0.5 + 0.866i)16-s + (−2.41 − 4.18i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0624 + 0.998i)3-s + (−0.249 − 0.433i)4-s − 1.07·5-s + (0.633 + 0.314i)6-s + (0.362 + 0.628i)7-s − 0.353·8-s + (−0.992 + 0.124i)9-s + (−0.379 + 0.656i)10-s + (0.867 + 1.50i)11-s + (0.416 − 0.276i)12-s + (0.638 + 1.10i)13-s + 0.512·14-s + (−0.0669 − 1.07i)15-s + (−0.125 + 0.216i)16-s + (−0.586 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999977 + 0.715236i\)
\(L(\frac12)\) \(\approx\) \(0.999977 + 0.715236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.108 - 1.72i)T \)
19 \( 1 + (-0.469 - 4.33i)T \)
good5 \( 1 + 2.39T + 5T^{2} \)
7 \( 1 + (-0.959 - 1.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.87 - 4.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.30 - 3.98i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.41 + 4.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.642 - 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.37T + 29T^{2} \)
31 \( 1 + (-2.04 + 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + (-1.98 + 3.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.84T + 47T^{2} \)
53 \( 1 + (-5.14 + 8.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.84T + 59T^{2} \)
61 \( 1 + 0.949T + 61T^{2} \)
67 \( 1 + (-5.01 - 8.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.68 + 4.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.26 - 12.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.74 + 8.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.715 + 1.23i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.59 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.40 - 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.66039893374572076225361882193, −11.13020614077820496779244232374, −9.731543125945616352477936088265, −9.261483837180156193758910508118, −8.187316171448119576837697489725, −6.88086429592581549298050650273, −5.43146006355997571561580255400, −4.26941252228140440808673791668, −3.87913653683909596318354981769, −2.16401336205258443327012709387, 0.805613583738713477282166031418, 3.19835264046375851181335195329, 4.13448071645395805534330332790, 5.72800367047598609830600118216, 6.55421477587092641982102298619, 7.59614028432745108102224398774, 8.235221292496341057981706646782, 8.914784236971933567130521505182, 11.05520130534128331277043114744, 11.21818229327035301514636852259

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