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Properties

Label 2-342-171.49-c1-0-4
Degree $2$
Conductor $342$
Sign $0.580 + 0.814i$
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  − 2-s + (−1.60 − 0.650i)3-s + 4-s + (−0.706 − 1.22i)5-s + (1.60 + 0.650i)6-s + (1.53 + 2.65i)7-s − 8-s + (2.15 + 2.08i)9-s + (0.706 + 1.22i)10-s + (−0.769 − 1.33i)11-s + (−1.60 − 0.650i)12-s + 1.33·13-s + (−1.53 − 2.65i)14-s + (0.337 + 2.42i)15-s + 16-s + (1.80 − 3.12i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.926 − 0.375i)3-s + 0.5·4-s + (−0.315 − 0.546i)5-s + (0.655 + 0.265i)6-s + (0.579 + 1.00i)7-s − 0.353·8-s + (0.717 + 0.696i)9-s + (0.223 + 0.386i)10-s + (−0.232 − 0.401i)11-s + (−0.463 − 0.187i)12-s + 0.368·13-s + (−0.409 − 0.710i)14-s + (0.0871 + 0.625i)15-s + 0.250·16-s + (0.437 − 0.757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623103 - 0.321252i\)
\(L(\frac12)\) \(\approx\) \(0.623103 - 0.321252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.60 + 0.650i)T \)
19 \( 1 + (2.00 + 3.86i)T \)
good5 \( 1 + (0.706 + 1.22i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.53 - 2.65i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.769 + 1.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 + (-1.80 + 3.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 + (-3.48 + 6.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.06 - 1.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 + (2.94 + 5.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (3.75 - 6.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.17 - 3.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.28 + 7.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.68 + 8.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 9.98T + 67T^{2} \)
71 \( 1 + (4.48 - 7.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.51 - 2.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 + (-7.47 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.08T + 97T^{2} \)
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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

−11.33664068776206620730313079267, −10.72547503865309040764870762154, −9.401186465127521552017965142025, −8.547028791146454648727911563878, −7.72848756197962461586267206108, −6.58511774768974815860113349487, −5.55240330835788476629767032253, −4.66041568866903708162108173644, −2.51172284328646110139340448916, −0.831398414285726950006597115709, 1.28534963248566593640912775751, 3.50671830638832329517115047329, 4.65466329699126332291932044829, 5.97952609260327864328226592013, 7.03434497354598628122930944102, 7.71256552007444984739507276657, 8.953123788720460193291793584956, 10.27271456862865353656334866773, 10.59095809317928027108540312053, 11.30180487351613507417775243756

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