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Properties

Label 2-342-171.49-c1-0-8
Degree $2$
Conductor $342$
Sign $0.248 - 0.968i$
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 + 1.73i·3-s + 4-s + (0.5 + 0.866i)5-s + 1.73i·6-s + (1.5 + 2.59i)7-s + 8-s − 2.99·9-s + (0.5 + 0.866i)10-s + (−2.5 − 4.33i)11-s + 1.73i·12-s + 2·13-s + (1.5 + 2.59i)14-s + (−1.49 + 0.866i)15-s + 16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.999i·3-s + 0.5·4-s + (0.223 + 0.387i)5-s + 0.707i·6-s + (0.566 + 0.981i)7-s + 0.353·8-s − 0.999·9-s + (0.158 + 0.273i)10-s + (−0.753 − 1.30i)11-s + 0.499i·12-s + 0.554·13-s + (0.400 + 0.694i)14-s + (−0.387 + 0.223i)15-s + 0.250·16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.248 - 0.968i$
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: \((2,\ 342,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62183 + 1.25822i\)
\(L(\frac12)\) \(\approx\) \(1.62183 + 1.25822i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.40190153891589955790280332913, −10.98849975913211296966802782715, −10.17883682805236280993077604193, −8.675696759428085970863012587246, −8.359397823576986422096068895207, −6.46408245962778035734504885865, −5.66613705133901932048182588163, −4.83023518400648843729895514383, −3.49291000160053885147360902085, −2.48587837204033969182971533948, 1.36304840853209317062136958931, 2.73264728677592100242283763532, 4.46882537847122203073230791800, 5.24841865649029091109051318919, 6.68439514025745018169231739367, 7.28443784383398991008619317301, 8.214253213284279590337826187077, 9.451686565765336558552569323693, 10.87556256137575613168263243321, 11.29600028459573387333540585573

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