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Properties

Label 2-342-171.65-c1-0-6
Degree $2$
Conductor $342$
Sign $0.888 + 0.458i$
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 + (−1.13 − 1.30i)3-s + (−0.499 + 0.866i)4-s + 2.55i·5-s + (−0.565 + 1.63i)6-s + (0.762 − 1.32i)7-s + 0.999·8-s + (−0.423 + 2.97i)9-s + (2.21 − 1.27i)10-s + (1.01 + 0.584i)11-s + (1.70 − 0.328i)12-s + (1.91 + 1.10i)13-s − 1.52·14-s + (3.33 − 2.89i)15-s + (−0.5 − 0.866i)16-s + (6.71 + 3.87i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.655 − 0.755i)3-s + (−0.249 + 0.433i)4-s + 1.14i·5-s + (−0.230 + 0.668i)6-s + (0.288 − 0.499i)7-s + 0.353·8-s + (−0.141 + 0.990i)9-s + (0.699 − 0.403i)10-s + (0.305 + 0.176i)11-s + (0.490 − 0.0949i)12-s + (0.531 + 0.306i)13-s − 0.407·14-s + (0.862 − 0.748i)15-s + (−0.125 − 0.216i)16-s + (1.62 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913825 - 0.221610i\)
\(L(\frac12)\) \(\approx\) \(0.913825 - 0.221610i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 + 1.30i)T \)
19 \( 1 + (1.14 + 4.20i)T \)
good5 \( 1 - 2.55iT - 5T^{2} \)
7 \( 1 + (-0.762 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.01 - 0.584i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.91 - 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.71 - 3.87i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.116 + 0.0674i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 + (-0.334 + 0.193i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.30iT - 37T^{2} \)
41 \( 1 + 6.75T + 41T^{2} \)
43 \( 1 + (-0.944 - 1.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.81iT - 47T^{2} \)
53 \( 1 + (-4.43 - 7.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.77T + 59T^{2} \)
61 \( 1 - 0.918T + 61T^{2} \)
67 \( 1 + (-6.16 - 3.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.47 - 7.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.36 - 9.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.72 + 5.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.4 + 7.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.33 + 5.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.462 - 0.267i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.28780649122449558731469706670, −10.72348862672185039095981658208, −10.01462940414185093646907055611, −8.556962827816351830946677967719, −7.52895527183316055788734195763, −6.82576004342839278204357514710, −5.76254448956284392175332252787, −4.18368739560249642105448708356, −2.79896883227496835170287228044, −1.30221387916918906562422172392, 1.02816941405563972573610788843, 3.65674671195760713914647526428, 5.06245235581553804712766565090, 5.45377432977765500855087154330, 6.60313320530877478045009135080, 8.142772365552411791950483380526, 8.719708786928641545462194249390, 9.726281284844201089970764171814, 10.37969668090342999353124470021, 11.81209758121381898296863661864

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