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Properties

Label 2-342-171.7-c1-0-10
Degree $2$
Conductor $342$
Sign $0.860 + 0.508i$
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 + (1.64 − 0.555i)3-s + 4-s + (0.467 − 0.809i)5-s + (−1.64 + 0.555i)6-s + (−0.568 + 0.984i)7-s − 8-s + (2.38 − 1.82i)9-s + (−0.467 + 0.809i)10-s + (0.0397 − 0.0689i)11-s + (1.64 − 0.555i)12-s + 6.63·13-s + (0.568 − 0.984i)14-s + (0.317 − 1.58i)15-s + 16-s + (−1.10 − 1.91i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.947 − 0.320i)3-s + 0.5·4-s + (0.209 − 0.362i)5-s + (−0.669 + 0.226i)6-s + (−0.214 + 0.371i)7-s − 0.353·8-s + (0.794 − 0.607i)9-s + (−0.147 + 0.256i)10-s + (0.0119 − 0.0207i)11-s + (0.473 − 0.160i)12-s + 1.84·13-s + (0.151 − 0.263i)14-s + (0.0819 − 0.410i)15-s + 0.250·16-s + (−0.268 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35319 - 0.369852i\)
\(L(\frac12)\) \(\approx\) \(1.35319 - 0.369852i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.555i)T \)
19 \( 1 + (4.19 + 1.18i)T \)
good5 \( 1 + (-0.467 + 0.809i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.568 - 0.984i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0397 + 0.0689i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.63T + 13T^{2} \)
17 \( 1 + (1.10 + 1.91i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + 0.138T + 23T^{2} \)
29 \( 1 + (2.94 + 5.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.71 - 2.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.58T + 37T^{2} \)
41 \( 1 + (-1.15 + 1.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.30T + 43T^{2} \)
47 \( 1 + (2.46 + 4.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.22 - 7.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.81 - 6.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.54 - 2.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (-6.23 - 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.81 + 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 + (4.76 - 8.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.11 - 7.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−11.30580519498486097845488361978, −10.38512827457703891274993801552, −9.123699173999103860521813260079, −8.877279308121263055178268326144, −7.935735210908647854651515509099, −6.80424548154576684301324700785, −5.87672795879863854013843749862, −4.08249273032525707957376038203, −2.77480612604847452987977479997, −1.39194158324092404421571949918, 1.72255572139679729280628099814, 3.18809136483010094464960225498, 4.20985801348023449368780212766, 6.07110239499125051635350702746, 6.95274991364464450378684919609, 8.196930222752478345612206706477, 8.690396829222491116917907234715, 9.710524922504502564574131417945, 10.61537013412289445022499010384, 11.07183357613512748438661528185

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