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Properties

Label 2-342-171.106-c1-0-14
Degree $2$
Conductor $342$
Sign $0.280 - 0.959i$
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  + (0.5 + 0.866i)2-s + (1.36 + 1.06i)3-s + (−0.499 + 0.866i)4-s + 1.41·5-s + (−0.238 + 1.71i)6-s + (1.53 − 2.65i)7-s − 0.999·8-s + (0.732 + 2.90i)9-s + (0.706 + 1.22i)10-s + (−0.769 + 1.33i)11-s + (−1.60 + 0.650i)12-s + (−0.665 + 1.15i)13-s + 3.06·14-s + (1.92 + 1.50i)15-s + (−0.5 − 0.866i)16-s + (1.80 − 3.12i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.788 + 0.614i)3-s + (−0.249 + 0.433i)4-s + 0.631·5-s + (−0.0975 + 0.700i)6-s + (0.579 − 1.00i)7-s − 0.353·8-s + (0.244 + 0.969i)9-s + (0.223 + 0.386i)10-s + (−0.232 + 0.401i)11-s + (−0.463 + 0.187i)12-s + (−0.184 + 0.319i)13-s + 0.819·14-s + (0.498 + 0.388i)15-s + (−0.125 − 0.216i)16-s + (0.437 − 0.757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.280 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73496 + 1.30112i\)
\(L(\frac12)\) \(\approx\) \(1.73496 + 1.30112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.36 - 1.06i)T \)
19 \( 1 + (2.00 + 3.86i)T \)
good5 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + (-1.53 + 2.65i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.769 - 1.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.665 - 1.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.80 + 3.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.49 - 4.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
31 \( 1 + (1.06 + 1.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 - 5.88T + 41T^{2} \)
43 \( 1 + (5.52 + 9.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.50T + 47T^{2} \)
53 \( 1 + (-2.17 - 3.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.56T + 59T^{2} \)
61 \( 1 + 9.37T + 61T^{2} \)
67 \( 1 + (-4.99 + 8.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.48 - 7.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.51 - 2.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.35 + 7.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.47 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.543 + 0.941i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.64748817982247397714463440399, −10.61310838872108345490298832697, −9.701625114743005958251437205201, −8.995244608112747028979819915491, −7.67700077347222280261567137023, −7.26577470384528522031529555627, −5.66224407113444609386621302093, −4.66379363586311679696067512845, −3.76497118239272979014586733011, −2.18476930517716502949326237534, 1.71460622788175736376289086773, 2.59762918617109212243354792481, 3.93856275744094497024293192989, 5.54568991162693083346238724676, 6.20057633264278287073804107679, 7.83652546164144714956132891113, 8.536357926686027743061357180800, 9.490022112542262123318519514851, 10.38313194491688100421614276320, 11.53645167455535040774870080674

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