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Properties

Label 2-342-171.41-c1-0-18
Degree $2$
Conductor $342$
Sign $-0.980 + 0.195i$
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 + (−1.28 − 1.16i)3-s + (−0.939 − 0.342i)4-s + (0.837 − 0.998i)5-s + (−1.37 + 1.05i)6-s + (1.63 − 2.82i)7-s + (−0.5 + 0.866i)8-s + (0.277 + 2.98i)9-s + (−0.837 − 0.998i)10-s − 2.05i·11-s + (0.803 + 1.53i)12-s + (−2.14 − 2.55i)13-s + (−2.49 − 2.09i)14-s + (−2.23 + 0.300i)15-s + (0.766 + 0.642i)16-s + (−1.31 + 1.57i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.739 − 0.673i)3-s + (−0.469 − 0.171i)4-s + (0.374 − 0.446i)5-s + (−0.559 + 0.431i)6-s + (0.616 − 1.06i)7-s + (−0.176 + 0.306i)8-s + (0.0923 + 0.995i)9-s + (−0.264 − 0.315i)10-s − 0.620i·11-s + (0.232 + 0.442i)12-s + (−0.595 − 0.709i)13-s + (−0.668 − 0.560i)14-s + (−0.577 + 0.0776i)15-s + (0.191 + 0.160i)16-s + (−0.319 + 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0982300 - 0.997085i\)
\(L(\frac12)\) \(\approx\) \(0.0982300 - 0.997085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 1.16i)T \)
19 \( 1 + (4.22 + 1.06i)T \)
good5 \( 1 + (-0.837 + 0.998i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.63 + 2.82i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.05iT - 11T^{2} \)
13 \( 1 + (2.14 + 2.55i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.31 - 1.57i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (2.16 - 5.95i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-3.03 - 1.10i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + 7.77iT - 31T^{2} \)
37 \( 1 + 8.22iT - 37T^{2} \)
41 \( 1 + (0.836 - 4.74i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-9.85 + 3.58i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.53 + 6.97i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-0.123 - 0.702i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.89 - 1.05i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.06 - 3.41i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-14.2 + 2.50i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.465 + 2.63i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.64 + 2.41i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-4.04 + 4.82i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-11.6 - 6.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.87 + 2.50i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.00 - 1.41i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.96853355942146590355042416543, −10.63530043725273617986454817668, −9.429584191390735332663514764358, −8.153378161971909949952210196595, −7.36026988294926957453849687917, −5.99326665249040549606871069581, −5.11126245635224751578012512777, −3.99151552485885060834684806977, −2.10452903955439717010734323210, −0.74960481555764260660677932885, 2.43804392252086648131393650959, 4.38021913749862570614001509944, 5.01895040084634953992374521148, 6.21165113287284354700950695244, 6.79101584483467100242998276848, 8.302261430055101472340737574603, 9.167365685344841739532835871065, 10.08579752757273259828823040268, 10.95602988207164745250831503690, 12.15350249813614239336067096170

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