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Properties

Label 2-342-171.142-c1-0-15
Degree $2$
Conductor $342$
Sign $-0.988 + 0.154i$
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  + (−0.939 − 0.342i)2-s + (−1.71 + 0.244i)3-s + (0.766 + 0.642i)4-s + (−0.582 − 3.30i)5-s + (1.69 + 0.356i)6-s + (1.43 + 2.48i)7-s + (−0.500 − 0.866i)8-s + (2.88 − 0.837i)9-s + (−0.582 + 3.30i)10-s − 4.22·11-s + (−1.47 − 0.915i)12-s + (0.835 − 4.74i)13-s + (−0.497 − 2.82i)14-s + (1.80 + 5.52i)15-s + (0.173 + 0.984i)16-s + (0.584 + 3.31i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.990 + 0.141i)3-s + (0.383 + 0.321i)4-s + (−0.260 − 1.47i)5-s + (0.691 + 0.145i)6-s + (0.541 + 0.938i)7-s + (−0.176 − 0.306i)8-s + (0.960 − 0.279i)9-s + (−0.184 + 1.04i)10-s − 1.27·11-s + (−0.424 − 0.264i)12-s + (0.231 − 1.31i)13-s + (−0.133 − 0.754i)14-s + (0.466 + 1.42i)15-s + (0.0434 + 0.246i)16-s + (0.141 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.988 + 0.154i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.988 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0197600 - 0.255043i\)
\(L(\frac12)\) \(\approx\) \(0.0197600 - 0.255043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (1.71 - 0.244i)T \)
19 \( 1 + (4.26 - 0.886i)T \)
good5 \( 1 + (0.582 + 3.30i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.43 - 2.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.22T + 11T^{2} \)
13 \( 1 + (-0.835 + 4.74i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.584 - 3.31i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.72 + 3.12i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.88 + 4.09i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + (-3.61 - 1.31i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-8.32 + 6.98i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.85 - 1.55i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.393 - 0.143i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (3.25 - 2.73i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.18 - 12.3i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.88 + 2.86i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.368 + 0.133i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.25 + 5.25i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.44 + 8.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.62 - 4.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.76 - 1.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.72 + 1.35i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.88687240952481890992713667980, −10.41592959515231716916537556685, −9.135998781983424011112521594863, −8.330303864562231222980690760274, −7.66068404402682310686678418254, −5.76355034952826794206228483041, −5.40075849897653134745490981888, −4.09219591551766216330917215811, −1.95395791467210755161623631199, −0.23765040234069985865176631159, 2.02464334867092419188133423106, 3.88737998787247521558158081347, 5.27721125373893957416276338236, 6.50648877722370487336537171263, 7.27099502294596935940253516302, 7.70759717990933193650580485620, 9.390513804662914798826399227294, 10.49062587862507457526384065601, 10.97103261542203019695645534778, 11.36177470485033309130266510003

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