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Properties

Label 2-342-171.25-c1-0-6
Degree $2$
Conductor $342$
Sign $0.998 + 0.0451i$
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 + (1.69 − 0.331i)3-s + (−0.939 + 0.342i)4-s + (−1.94 + 1.62i)5-s + (−0.621 − 1.61i)6-s + (2.14 + 3.71i)7-s + (0.5 + 0.866i)8-s + (2.77 − 1.12i)9-s + (1.94 + 1.62i)10-s − 0.213·11-s + (−1.48 + 0.893i)12-s + (2.37 + 1.99i)13-s + (3.28 − 2.75i)14-s + (−2.75 + 3.41i)15-s + (0.766 − 0.642i)16-s + (1.85 − 1.55i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.981 − 0.191i)3-s + (−0.469 + 0.171i)4-s + (−0.867 + 0.728i)5-s + (−0.253 − 0.659i)6-s + (0.810 + 1.40i)7-s + (0.176 + 0.306i)8-s + (0.926 − 0.376i)9-s + (0.613 + 0.514i)10-s − 0.0644·11-s + (−0.428 + 0.257i)12-s + (0.659 + 0.553i)13-s + (0.878 − 0.737i)14-s + (−0.712 + 0.880i)15-s + (0.191 − 0.160i)16-s + (0.449 − 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.998 + 0.0451i$
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: \((2,\ 342,\ (\ :1/2),\ 0.998 + 0.0451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57236 - 0.0355033i\)
\(L(\frac12)\) \(\approx\) \(1.57236 - 0.0355033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.331i)T \)
19 \( 1 + (-1.04 + 4.23i)T \)
good5 \( 1 + (1.94 - 1.62i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.14 - 3.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.213T + 11T^{2} \)
13 \( 1 + (-2.37 - 1.99i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.85 + 1.55i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.959 + 0.349i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (8.16 - 2.97i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 - 8.11T + 37T^{2} \)
41 \( 1 + (1.39 + 7.92i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.26 - 1.55i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.34 + 0.488i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-0.236 + 1.33i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-5.30 - 1.93i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.38 - 2.83i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.65 + 9.39i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.57 + 14.6i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (3.62 + 1.31i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (4.28 - 3.59i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.29 - 5.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (10.9 - 3.98i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.45 + 13.9i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.39601353092980820213309024094, −10.93713687834711883178186196448, −9.317282478174402613590217264047, −8.935100732657062956401257196407, −7.88253069175094042377310806689, −7.12295425832358880225650702543, −5.43864354243799611670324299887, −4.01298940719641907696522874031, −3.00149208793600184581542355854, −1.92004006434387316186733637322, 1.25310357321796939815663987014, 3.76528339126638838938416554542, 4.20843315712322428187595541572, 5.52426726341830762451262649680, 7.27803891753201060573724885443, 7.86826426980192396649882241722, 8.333664607446665648201306693360, 9.481903910952656616557689333251, 10.46103043969326950394180128765, 11.37534470044077954522932412025

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