↑

Properties

Label 2-336-16.13-c1-0-23
Degree $2$
Conductor $336$
Sign $-0.921 + 0.389i$
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  + (0.783 − 1.17i)2-s + (−0.707 − 0.707i)3-s + (−0.771 − 1.84i)4-s + (0.134 − 0.134i)5-s + (−1.38 + 0.278i)6-s − i·7-s + (−2.77 − 0.537i)8-s + 1.00i·9-s + (−0.0527 − 0.262i)10-s + (2.78 − 2.78i)11-s + (−0.759 + 1.85i)12-s + (−3.77 − 3.77i)13-s + (−1.17 − 0.783i)14-s − 0.189·15-s + (−2.80 + 2.84i)16-s − 6.01·17-s + ⋯
L(s)  = 1  + (0.554 − 0.832i)2-s + (−0.408 − 0.408i)3-s + (−0.385 − 0.922i)4-s + (0.0599 − 0.0599i)5-s + (−0.566 + 0.113i)6-s − 0.377i·7-s + (−0.981 − 0.190i)8-s + 0.333i·9-s + (−0.0166 − 0.0831i)10-s + (0.840 − 0.840i)11-s + (−0.219 + 0.534i)12-s + (−1.04 − 1.04i)13-s + (−0.314 − 0.209i)14-s − 0.0489·15-s + (−0.702 + 0.711i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.921 + 0.389i$
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: \((2,\ 336,\ (\ :1/2),\ -0.921 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257926 - 1.27382i\)
\(L(\frac12)\) \(\approx\) \(0.257926 - 1.27382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.783 + 1.17i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-0.134 + 0.134i)T - 5iT^{2} \)
11 \( 1 + (-2.78 + 2.78i)T - 11iT^{2} \)
13 \( 1 + (3.77 + 3.77i)T + 13iT^{2} \)
17 \( 1 + 6.01T + 17T^{2} \)
19 \( 1 + (-3.25 - 3.25i)T + 19iT^{2} \)
23 \( 1 + 3.03iT - 23T^{2} \)
29 \( 1 + (-2.55 - 2.55i)T + 29iT^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (-6.76 + 6.76i)T - 37iT^{2} \)
41 \( 1 + 7.47iT - 41T^{2} \)
43 \( 1 + (-6.64 + 6.64i)T - 43iT^{2} \)
47 \( 1 + 6.53T + 47T^{2} \)
53 \( 1 + (-0.936 + 0.936i)T - 53iT^{2} \)
59 \( 1 + (0.924 - 0.924i)T - 59iT^{2} \)
61 \( 1 + (-4.61 - 4.61i)T + 61iT^{2} \)
67 \( 1 + (-7.37 - 7.37i)T + 67iT^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 - 4.66T + 79T^{2} \)
83 \( 1 + (0.636 + 0.636i)T + 83iT^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + 9.78T + 97T^{2} \)
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

−11.23641536060626828096647978746, −10.51698111101845047311894923871, −9.547801763808466162987452572347, −8.467314248880018011799341468017, −7.09686028309962793179405859374, −6.05443085416912292974542562052, −5.09386797527142551868835701711, −3.89902828655730425234637681674, −2.53406660453667324057119922466, −0.830420926934566288300599966069, 2.62693808301076452211713336389, 4.46398416581571842972554133916, 4.71220366004262297211888542396, 6.35767628504678681860963125441, 6.76974453099885217104460090900, 8.050474856654535637353482478404, 9.303892865957171984147950233858, 9.721361462854832919680678928244, 11.56182767629458912282882186408, 11.78296722516097378251025280570

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