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Properties

Label 2-342-171.25-c1-0-5
Degree $2$
Conductor $342$
Sign $0.936 - 0.349i$
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  + (−0.173 − 0.984i)2-s + (0.936 + 1.45i)3-s + (−0.939 + 0.342i)4-s + (−0.0485 + 0.0406i)5-s + (1.27 − 1.17i)6-s + (0.0578 + 0.100i)7-s + (0.5 + 0.866i)8-s + (−1.24 + 2.72i)9-s + (0.0485 + 0.0406i)10-s + 3.65·11-s + (−1.37 − 1.04i)12-s + (2.74 + 2.30i)13-s + (0.0886 − 0.0743i)14-s + (−0.104 − 0.0325i)15-s + (0.766 − 0.642i)16-s + (1.24 − 1.04i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.540 + 0.841i)3-s + (−0.469 + 0.171i)4-s + (−0.0216 + 0.0182i)5-s + (0.519 − 0.479i)6-s + (0.0218 + 0.0378i)7-s + (0.176 + 0.306i)8-s + (−0.414 + 0.909i)9-s + (0.0153 + 0.0128i)10-s + 1.10·11-s + (−0.397 − 0.302i)12-s + (0.760 + 0.638i)13-s + (0.0236 − 0.0198i)14-s + (−0.0270 − 0.00840i)15-s + (0.191 − 0.160i)16-s + (0.302 − 0.253i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44150 + 0.260032i\)
\(L(\frac12)\) \(\approx\) \(1.44150 + 0.260032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.936 - 1.45i)T \)
19 \( 1 + (2.44 - 3.60i)T \)
good5 \( 1 + (0.0485 - 0.0406i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.0578 - 0.100i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
13 \( 1 + (-2.74 - 2.30i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.24 + 1.04i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.81 + 0.661i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-3.24 + 1.17i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 + 2.48T + 37T^{2} \)
41 \( 1 + (1.80 + 10.2i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (9.09 + 3.31i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-4.48 + 1.63i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.55 + 8.83i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (8.27 + 3.01i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (4.19 + 3.52i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.728 + 4.12i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.68 - 9.53i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-3.78 - 1.37i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.40 - 7.89i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.635 - 1.10i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.96 + 0.713i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.854 - 4.84i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.46317808731958021149165693216, −10.60088721887438597760353703800, −9.749192528489740181976396687560, −8.925238104080296282206539101982, −8.291772400430169186369649489254, −6.82656887897300175680008709940, −5.39773084055253601588849626038, −4.12035926596373065026146496054, −3.43387382379126504367787534105, −1.82608748076256442858746622411, 1.20586423633967195975581351486, 3.08480833251098951889388570464, 4.43874806904225612414634241348, 6.06492068727163427855684671561, 6.62130095412128046557212738260, 7.71962686496527528144899727913, 8.548832959904564736056441157969, 9.209045817117302042269542056670, 10.43634952253724044696252077548, 11.65404379986076592479728685830

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