↑

Properties

Label 2-338-13.4-c1-0-5
Degree $2$
Conductor $338$
Sign $0.265 - 0.964i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + 3i·5-s + (0.866 − 0.499i)6-s + (−2.59 + 1.5i)7-s + 0.999i·8-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)10-s + 0.999·12-s − 3·14-s + (2.59 + 1.5i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + 2i·18-s + (5.19 − 3i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 1.34i·5-s + (0.353 − 0.204i)6-s + (−0.981 + 0.566i)7-s + 0.353i·8-s + (0.333 + 0.577i)9-s + (−0.474 + 0.821i)10-s + 0.288·12-s − 0.801·14-s + (0.670 + 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + 0.471i·18-s + (1.19 − 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48621 + 1.13286i\)
\(L(\frac12)\) \(\approx\) \(1.48621 + 1.13286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.19 + 3i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.9 - 7.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 6i)T + (48.5 - 84.0i)T^{2} \)
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

−11.83762648043275240398486337476, −10.93680261986639105718194087181, −9.961775113856470084607581699581, −8.842600589688149352842128921980, −7.50128335923766245257959346550, −6.93362396641257135021314190712, −6.15194600084190825966642779241, −4.81195262289793544546799267651, −3.14309886222369328409028049252, −2.55002698192798751876148732271, 1.18223836985414581431463868065, 3.32137303866028164850601828116, 4.09173494102892783268122105548, 5.15894016117183705483374519158, 6.27300578741335073784905080307, 7.51123789265954325323175874371, 8.897721293488655582199789464590, 9.571018817252374331873220709490, 10.25148372261760366859462675657, 11.53976802188354313074633725160

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