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Properties

Label 2-342-171.106-c1-0-9
Degree $2$
Conductor $342$
Sign $0.276 - 0.960i$
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 + (1.73 + 0.0330i)3-s + (−0.499 + 0.866i)4-s + 0.730·5-s + (0.837 + 1.51i)6-s + (−1.79 + 3.10i)7-s − 0.999·8-s + (2.99 + 0.114i)9-s + (0.365 + 0.633i)10-s + (0.211 − 0.365i)11-s + (−0.894 + 1.48i)12-s + (0.610 − 1.05i)13-s − 3.58·14-s + (1.26 + 0.0241i)15-s + (−0.5 − 0.866i)16-s + (−2.50 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.999 + 0.0190i)3-s + (−0.249 + 0.433i)4-s + 0.326·5-s + (0.341 + 0.619i)6-s + (−0.676 + 1.17i)7-s − 0.353·8-s + (0.999 + 0.0381i)9-s + (0.115 + 0.200i)10-s + (0.0637 − 0.110i)11-s + (−0.258 + 0.428i)12-s + (0.169 − 0.293i)13-s − 0.956·14-s + (0.326 + 0.00624i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.276 - 0.960i$
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: \((2,\ 342,\ (\ :1/2),\ 0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67400 + 1.25980i\)
\(L(\frac12)\) \(\approx\) \(1.67400 + 1.25980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0330i)T \)
19 \( 1 + (-3.13 + 3.02i)T \)
good5 \( 1 - 0.730T + 5T^{2} \)
7 \( 1 + (1.79 - 3.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.211 + 0.365i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.610 + 1.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.50 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.96 + 6.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.02T + 29T^{2} \)
31 \( 1 + (3.60 + 6.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 4.22T + 41T^{2} \)
43 \( 1 + (-0.186 - 0.323i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.54T + 47T^{2} \)
53 \( 1 + (-1.03 - 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.14T + 59T^{2} \)
61 \( 1 + 8.81T + 61T^{2} \)
67 \( 1 + (0.515 - 0.893i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.21 + 9.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.49 + 7.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.82 + 4.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.43 - 7.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.11 - 3.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.51 - 7.82i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.09404773141367155518003874608, −10.62504810977320982576888723654, −9.492559591700010191561674003600, −8.836203651511087788075907191233, −8.104523460703545265476477595053, −6.80690606817687570874941950458, −6.01804617152415571780916963633, −4.75125731760181784890692041251, −3.38518216741231248256348576686, −2.35145030923272929670926083750, 1.47046958376774894541271652622, 3.06328494202747529548304610360, 3.84904389053271463071166305945, 5.08415633795925415665720244355, 6.71801225104087089436991744396, 7.42711425652118316545149071903, 8.793595589025689740674771503892, 9.701247548390542637879251827145, 10.17697371151172357559728925789, 11.31599112535584726892517093228

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