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Properties

Label 2-342-171.106-c1-0-19
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: $\chi_{342} (277, \cdot )$
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

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

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