Properties

Label 2-342-171.86-c1-0-3
Degree $2$
Conductor $342$
Sign $0.990 - 0.135i$
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 + (−1.72 − 0.0916i)3-s + (0.173 − 0.984i)4-s + (2.16 + 2.58i)5-s + (−1.38 + 1.04i)6-s − 1.76·7-s + (−0.500 − 0.866i)8-s + (2.98 + 0.317i)9-s + (3.32 + 0.586i)10-s + (2.40 + 1.38i)11-s + (−0.390 + 1.68i)12-s + (1.63 + 4.48i)13-s + (−1.35 + 1.13i)14-s + (−3.51 − 4.67i)15-s + (−0.939 − 0.342i)16-s + (2.88 − 0.508i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.998 − 0.0529i)3-s + (0.0868 − 0.492i)4-s + (0.970 + 1.15i)5-s + (−0.564 + 0.425i)6-s − 0.668·7-s + (−0.176 − 0.306i)8-s + (0.994 + 0.105i)9-s + (1.05 + 0.185i)10-s + (0.724 + 0.418i)11-s + (−0.112 + 0.487i)12-s + (0.452 + 1.24i)13-s + (−0.361 + 0.303i)14-s + (−0.907 − 1.20i)15-s + (−0.234 − 0.0855i)16-s + (0.698 − 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.990 - 0.135i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50253 + 0.102163i\)
\(L(\frac12)\) \(\approx\) \(1.50253 + 0.102163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.0916i)T \)
19 \( 1 + (-1.30 + 4.15i)T \)
good5 \( 1 + (-2.16 - 2.58i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.63 - 4.48i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.88 + 0.508i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.02 - 0.357i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (4.01 - 1.46i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.11 + 2.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.17iT - 37T^{2} \)
41 \( 1 + (2.10 + 11.9i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.03 - 11.5i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.14 + 3.14i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.81 + 1.75i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (12.0 + 4.39i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.03 + 5.90i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.00676 + 0.00806i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (7.49 + 2.72i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.25 + 1.05i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.15 - 11.4i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + 1.23iT - 83T^{2} \)
89 \( 1 + (0.898 + 0.753i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (5.26 + 6.27i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.46583454545859718426105793929, −10.83686352974089420805357199152, −9.823408224648888213706833739519, −9.380857111674871994979907190854, −7.07547537874069360337554116933, −6.57277605996937294171844810374, −5.81789620105213523982710519973, −4.54165047569467013350991645190, −3.19831239383862252284770116018, −1.70800008436004034202028049263, 1.18973981771471909966494569185, 3.49749275223315169852670489388, 4.82257443227297582645413482263, 5.92219887714200930795024231058, 5.98921257504912795051866361845, 7.52362233571050782444662651518, 8.751910196564566774422658317980, 9.718370263612632461642665915803, 10.52922967924230617606969768590, 11.82516827466814415013671118116

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