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Properties

Label 2-342-171.106-c1-0-12
Degree $2$
Conductor $342$
Sign $0.929 + 0.368i$
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 + (−1.70 − 0.277i)3-s + (−0.499 + 0.866i)4-s − 1.57·5-s + (−0.614 − 1.61i)6-s + (2.31 − 4.01i)7-s − 0.999·8-s + (2.84 + 0.948i)9-s + (−0.789 − 1.36i)10-s + (−1.83 + 3.18i)11-s + (1.09 − 1.34i)12-s + (2.78 − 4.82i)13-s + 4.63·14-s + (2.70 + 0.438i)15-s + (−0.5 − 0.866i)16-s + (3.65 − 6.32i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.987 − 0.160i)3-s + (−0.249 + 0.433i)4-s − 0.706·5-s + (−0.250 − 0.661i)6-s + (0.875 − 1.51i)7-s − 0.353·8-s + (0.948 + 0.316i)9-s + (−0.249 − 0.432i)10-s + (−0.554 + 0.959i)11-s + (0.316 − 0.387i)12-s + (0.772 − 1.33i)13-s + 1.23·14-s + (0.697 + 0.113i)15-s + (−0.125 − 0.216i)16-s + (0.886 − 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.929 + 0.368i$
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: \((2,\ 342,\ (\ :1/2),\ 0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04177 - 0.198812i\)
\(L(\frac12)\) \(\approx\) \(1.04177 - 0.198812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.277i)T \)
19 \( 1 + (-3.55 - 2.52i)T \)
good5 \( 1 + 1.57T + 5T^{2} \)
7 \( 1 + (-2.31 + 4.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.83 - 3.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.78 + 4.82i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.65 + 6.32i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.05 + 1.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 + (-0.587 - 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.16T + 37T^{2} \)
41 \( 1 + 4.49T + 41T^{2} \)
43 \( 1 + (4.24 + 7.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.75T + 47T^{2} \)
53 \( 1 + (-1.69 - 2.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.98T + 59T^{2} \)
61 \( 1 - 3.18T + 61T^{2} \)
67 \( 1 + (6.29 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.03 + 6.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.45 + 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.99 - 5.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.85 - 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.00 - 5.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.97 - 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.62276750862493281412603382407, −10.57131886587941353489252310681, −9.991873046346221568544622484443, −7.964139568254839604280606731163, −7.64457354431576746308988204501, −6.83067157511276857137685824314, −5.33522432173853566139042832875, −4.74232768669163869386629554618, −3.57380977780404191949341451849, −0.856225498344237648363145308103, 1.57728558164403108224564015705, 3.39774953602993524156598822226, 4.63295360848891995064971960060, 5.56790178592208733081478273904, 6.32459673422612962718958394890, 8.011230735904547031160827166244, 8.796772855816764003053739855272, 9.981406502107514098376140265951, 11.24265711135106769162424682177, 11.44996395952576087236306848754

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