↑

Properties

Label 2-342-171.106-c1-0-13
Degree $2$
Conductor $342$
Sign $0.980 + 0.194i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.378 + 1.69i)3-s + (−0.499 + 0.866i)4-s + 2.54·5-s + (1.27 − 1.17i)6-s + (2.28 − 3.95i)7-s + 0.999·8-s + (−2.71 + 1.28i)9-s + (−1.27 − 2.20i)10-s + (0.818 − 1.41i)11-s + (−1.65 − 0.517i)12-s + (−2.19 + 3.80i)13-s − 4.56·14-s + (0.965 + 4.30i)15-s + (−0.5 − 0.866i)16-s + (2.60 − 4.50i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.218 + 0.975i)3-s + (−0.249 + 0.433i)4-s + 1.13·5-s + (0.520 − 0.478i)6-s + (0.863 − 1.49i)7-s + 0.353·8-s + (−0.904 + 0.426i)9-s + (−0.402 − 0.697i)10-s + (0.246 − 0.427i)11-s + (−0.477 − 0.149i)12-s + (−0.609 + 1.05i)13-s − 1.22·14-s + (0.249 + 1.11i)15-s + (−0.125 − 0.216i)16-s + (0.631 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.980 + 0.194i$
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: \((2,\ 342,\ (\ :1/2),\ 0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46461 - 0.143595i\)
\(L(\frac12)\) \(\approx\) \(1.46461 - 0.143595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.378 - 1.69i)T \)
19 \( 1 + (-1.58 - 4.05i)T \)
good5 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 + (-2.28 + 3.95i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.818 + 1.41i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.19 - 3.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.60 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.796 + 1.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + (-3.29 - 5.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.93T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 + (-4.76 - 8.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.51T + 47T^{2} \)
53 \( 1 + (1.04 + 1.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + (-1.35 + 2.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.95 - 3.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.127 + 0.221i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.22 + 7.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.74 - 4.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.37 + 7.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.68 + 2.91i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.27526018036363812269759614793, −10.21278228248687320366554756203, −10.01999864583066684983321089268, −8.998738836061607805831686914659, −7.986983578147227157162254293495, −6.78827416389121285826524453904, −5.18786277915565233824964243761, −4.38648720925734801201798464600, −3.11598660654595810189042019037, −1.51766955540020902093066305521, 1.62437191640598828880788730557, 2.65260664311114797692445215375, 5.22108070999654272029013756070, 5.71831869521291712175014993446, 6.72491558826126355483183706731, 7.905161845225128536839366149135, 8.583924094324580332027721455375, 9.431516272568874662976350379545, 10.42497707780181971026091056566, 11.81814314114845879438038750954

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