↑

Properties

Label 2-336-16.13-c1-0-2
Degree $2$
Conductor $336$
Sign $-0.998 + 0.0463i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.884 + 1.10i)2-s + (0.707 + 0.707i)3-s + (−0.435 − 1.95i)4-s + (−1.84 + 1.84i)5-s + (−1.40 + 0.154i)6-s + i·7-s + (2.53 + 1.24i)8-s + 1.00i·9-s + (−0.404 − 3.67i)10-s + (−1.28 + 1.28i)11-s + (1.07 − 1.68i)12-s + (0.573 + 0.573i)13-s + (−1.10 − 0.884i)14-s − 2.61·15-s + (−3.62 + 1.70i)16-s − 3.76·17-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + (0.408 + 0.408i)3-s + (−0.217 − 0.975i)4-s + (−0.826 + 0.826i)5-s + (−0.573 + 0.0632i)6-s + 0.377i·7-s + (0.897 + 0.440i)8-s + 0.333i·9-s + (−0.127 − 1.16i)10-s + (−0.386 + 0.386i)11-s + (0.309 − 0.487i)12-s + (0.159 + 0.159i)13-s + (−0.294 − 0.236i)14-s − 0.674·15-s + (−0.905 + 0.425i)16-s − 0.913·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.998 + 0.0463i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.998 + 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0145696 - 0.628189i\)
\(L(\frac12)\) \(\approx\) \(0.0145696 - 0.628189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.884 - 1.10i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (1.84 - 1.84i)T - 5iT^{2} \)
11 \( 1 + (1.28 - 1.28i)T - 11iT^{2} \)
13 \( 1 + (-0.573 - 0.573i)T + 13iT^{2} \)
17 \( 1 + 3.76T + 17T^{2} \)
19 \( 1 + (5.58 + 5.58i)T + 19iT^{2} \)
23 \( 1 - 8.77iT - 23T^{2} \)
29 \( 1 + (2.45 + 2.45i)T + 29iT^{2} \)
31 \( 1 - 7.44T + 31T^{2} \)
37 \( 1 + (8.07 - 8.07i)T - 37iT^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
43 \( 1 + (-2.55 + 2.55i)T - 43iT^{2} \)
47 \( 1 - 8.68T + 47T^{2} \)
53 \( 1 + (1.03 - 1.03i)T - 53iT^{2} \)
59 \( 1 + (-4.49 + 4.49i)T - 59iT^{2} \)
61 \( 1 + (-1.77 - 1.77i)T + 61iT^{2} \)
67 \( 1 + (-9.10 - 9.10i)T + 67iT^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 - 1.67iT - 73T^{2} \)
79 \( 1 - 8.98T + 79T^{2} \)
83 \( 1 + (-5.03 - 5.03i)T + 83iT^{2} \)
89 \( 1 - 3.53iT - 89T^{2} \)
97 \( 1 - 7.92T + 97T^{2} \)
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

−11.60642358993240393800975230864, −10.95449784342608316132609119460, −10.05504381834745628032698132607, −9.053490176180789298515285212893, −8.290815367414133498535609472824, −7.29960324096101769345537320754, −6.57410275978391189260263869133, −5.15766007544616203819015716838, −4.01148193834962322395287606772, −2.40183850493048602316399004424, 0.50280457986850647482389615673, 2.21509419593466431344329731896, 3.73577173690139463570103724355, 4.57541148959475005946229912990, 6.49227735731537263294946281449, 7.73645779193675093596406201374, 8.427956667085582315647479346603, 8.916995193914520563964297695142, 10.36370773140976812868583761030, 10.93367552092021654015329650070

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