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Properties

Label 2-336-48.11-c1-0-14
Degree $2$
Conductor $336$
Sign $-0.644 - 0.764i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
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  + (1.31 + 0.508i)2-s + (−0.945 + 1.45i)3-s + (1.48 + 1.34i)4-s + (−1.69 + 1.69i)5-s + (−1.98 + 1.43i)6-s + 7-s + (1.27 + 2.52i)8-s + (−1.21 − 2.74i)9-s + (−3.09 + 1.37i)10-s + (−0.445 − 0.445i)11-s + (−3.34 + 0.883i)12-s + (−1.51 + 1.51i)13-s + (1.31 + 0.508i)14-s + (−0.856 − 4.06i)15-s + (0.401 + 3.97i)16-s + 1.39i·17-s + ⋯
L(s)  = 1  + (0.933 + 0.359i)2-s + (−0.546 + 0.837i)3-s + (0.741 + 0.670i)4-s + (−0.758 + 0.758i)5-s + (−0.810 + 0.585i)6-s + 0.377·7-s + (0.451 + 0.892i)8-s + (−0.403 − 0.914i)9-s + (−0.980 + 0.435i)10-s + (−0.134 − 0.134i)11-s + (−0.966 + 0.255i)12-s + (−0.419 + 0.419i)13-s + (0.352 + 0.135i)14-s + (−0.221 − 1.04i)15-s + (0.100 + 0.994i)16-s + 0.337i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.644 - 0.764i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701133 + 1.50920i\)
\(L(\frac12)\) \(\approx\) \(0.701133 + 1.50920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.508i)T \)
3 \( 1 + (0.945 - 1.45i)T \)
7 \( 1 - T \)
good5 \( 1 + (1.69 - 1.69i)T - 5iT^{2} \)
11 \( 1 + (0.445 + 0.445i)T + 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 - 1.39iT - 17T^{2} \)
19 \( 1 + (-0.965 - 0.965i)T + 19iT^{2} \)
23 \( 1 + 6.04iT - 23T^{2} \)
29 \( 1 + (-4.93 - 4.93i)T + 29iT^{2} \)
31 \( 1 + 0.470iT - 31T^{2} \)
37 \( 1 + (-1.47 - 1.47i)T + 37iT^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 + (-8.97 + 8.97i)T - 43iT^{2} \)
47 \( 1 - 4.84T + 47T^{2} \)
53 \( 1 + (3.42 - 3.42i)T - 53iT^{2} \)
59 \( 1 + (6.61 + 6.61i)T + 59iT^{2} \)
61 \( 1 + (8.04 - 8.04i)T - 61iT^{2} \)
67 \( 1 + (-4.38 - 4.38i)T + 67iT^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 16.5iT - 79T^{2} \)
83 \( 1 + (-9.24 + 9.24i)T - 83iT^{2} \)
89 \( 1 + 5.25T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−11.99735796401237954138296324739, −10.88017870089548752760899431268, −10.64889831310417496529145474262, −9.037214845815610853693882280401, −7.85565352535083441310853857911, −6.88920769229430069022041235287, −5.93750691805788650447431390590, −4.77060956584110735612771425955, −3.97586094984109183390727689192, −2.83142971021313648380439138345, 0.991497538548138920307163668864, 2.61827036585114149356139452218, 4.30440847840312490372480750459, 5.14497939021118678624678940811, 6.10353632783019505377936801695, 7.40983007549586041805626965871, 7.962229661235342482934738757209, 9.502648438172935125910397167230, 10.80076070092274977514572527949, 11.53250481209917805788243515501

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