↑

Properties

Label 2-336-112.101-c2-0-11
Degree $2$
Conductor $336$
Sign $0.966 - 0.254i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.682 − 1.87i)2-s + (−1.67 + 0.448i)3-s + (−3.06 − 2.56i)4-s + (−5.43 − 1.45i)5-s + (−0.298 + 3.45i)6-s + (3.99 + 5.74i)7-s + (−6.91 + 4.01i)8-s + (2.59 − 1.50i)9-s + (−6.44 + 9.21i)10-s + (−0.217 + 0.0582i)11-s + (6.28 + 2.91i)12-s + (−6.38 − 6.38i)13-s + (13.5 − 3.58i)14-s + 9.73·15-s + (2.83 + 15.7i)16-s + (27.4 + 15.8i)17-s + ⋯
L(s)  = 1  + (0.341 − 0.939i)2-s + (−0.557 + 0.149i)3-s + (−0.767 − 0.641i)4-s + (−1.08 − 0.291i)5-s + (−0.0498 + 0.575i)6-s + (0.570 + 0.821i)7-s + (−0.864 + 0.502i)8-s + (0.288 − 0.166i)9-s + (−0.644 + 0.921i)10-s + (−0.0197 + 0.00529i)11-s + (0.523 + 0.243i)12-s + (−0.491 − 0.491i)13-s + (0.966 − 0.256i)14-s + 0.649·15-s + (0.177 + 0.984i)16-s + (1.61 + 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.966 - 0.254i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.966 - 0.254i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.953676 + 0.123563i\)
\(L(\frac12)\) \(\approx\) \(0.953676 + 0.123563i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.682 + 1.87i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (-3.99 - 5.74i)T \)
good5 \( 1 + (5.43 + 1.45i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (0.217 - 0.0582i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (6.38 + 6.38i)T + 169iT^{2} \)
17 \( 1 + (-27.4 - 15.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.66 - 24.8i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-12.6 + 7.30i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-21.2 + 21.2i)T - 841iT^{2} \)
31 \( 1 + (-31.3 - 18.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (13.0 - 48.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 52.6T + 1.68e3T^{2} \)
43 \( 1 + (-18.6 - 18.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-6.34 + 3.66i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (32.3 - 8.66i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-13.9 - 52.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (26.4 - 98.6i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (29.2 + 109. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 54.0iT - 5.04e3T^{2} \)
73 \( 1 + (71.4 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-34.1 - 59.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-65.3 - 65.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-31.3 - 54.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 37.0iT - 9.40e3T^{2} \)
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

−11.68921524101370331605393095549, −10.52228942206345666259277343342, −9.939302407129990094422844452790, −8.464185264949971030700770417165, −7.971648417263858476588042954068, −6.10787898699920156020101717822, −5.17757408621500832248789286305, −4.26099691718638864662806902858, −3.07846613388749155493983640300, −1.27858961795126821360827328713, 0.50482227178564578412594580925, 3.31345273241898764303067846365, 4.49966455634076815425814086108, 5.20786783214089182989231991071, 6.74762094743103877898443089930, 7.34951054014435684793948374911, 7.990767502122133028150475243511, 9.270762134478096197895537799274, 10.49183725933680284449279356363, 11.59785063414080270802442127661

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