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Properties

Label 2-342-57.32-c1-0-0
Degree $2$
Conductor $342$
Sign $-0.553 - 0.832i$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−3.92 + 0.692i)5-s + (−2.50 + 4.33i)7-s + (−0.500 − 0.866i)8-s + (−2.56 + 3.05i)10-s + (−2.02 + 1.16i)11-s + (−0.706 − 1.94i)13-s + (0.869 + 4.93i)14-s + (−0.939 − 0.342i)16-s + (−2.08 − 2.48i)17-s + (3.25 + 2.89i)19-s + 3.98i·20-s + (−0.799 + 2.19i)22-s + (0.0315 + 0.00556i)23-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.75 + 0.309i)5-s + (−0.946 + 1.63i)7-s + (−0.176 − 0.306i)8-s + (−0.810 + 0.965i)10-s + (−0.610 + 0.352i)11-s + (−0.195 − 0.538i)13-s + (0.232 + 1.31i)14-s + (−0.234 − 0.0855i)16-s + (−0.505 − 0.601i)17-s + (0.747 + 0.664i)19-s + 0.891i·20-s + (−0.170 + 0.468i)22-s + (0.00658 + 0.00116i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209042 + 0.389886i\)
\(L(\frac12)\) \(\approx\) \(0.209042 + 0.389886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 \)
19 \( 1 + (-3.25 - 2.89i)T \)
good5 \( 1 + (3.92 - 0.692i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (2.50 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.706 + 1.94i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.08 + 2.48i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.0315 - 0.00556i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.785 - 0.658i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.59 + 3.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.00iT - 37T^{2} \)
41 \( 1 + (5.39 + 1.96i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.852 - 4.83i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.996 + 1.18i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (2.46 - 13.9i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.23 - 1.03i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.304 + 1.72i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.98 - 3.55i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.43 - 13.7i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (5.33 + 1.94i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-3.77 + 10.3i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.97 - 4.02i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.45 + 2.71i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (7.98 + 9.51i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.99087976282749584369050140244, −11.25693254167858039436655502938, −10.13512863371106024222858391085, −9.103017794598727787985777764799, −8.053798864192315350899964426635, −7.07459649357043129006648727122, −5.83464118342527877660212363241, −4.77415156641521957799257294820, −3.41208014319252267171414214593, −2.69661245586312919057383660642, 0.24847122642872433522264195638, 3.38794722228355345505919801137, 3.97218883703802029892418598649, 4.97297823026251758659496463758, 6.68514450514214019673275756321, 7.30328030595302068434173509422, 8.029736735300069824193155366684, 9.175485654675649262219230381436, 10.58857151300035134309830053874, 11.23602944319734359853280558504

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