↑

Properties

Label 2-338-13.10-c1-0-5
Degree $2$
Conductor $338$
Sign $0.711 + 0.702i$
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 + (0.499 − 0.866i)4-s + i·5-s + (−3.46 − 2i)7-s + 0.999i·8-s + (1.5 − 2.59i)9-s + (−0.5 − 0.866i)10-s + (3.46 − 2i)11-s + 3.99·14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + 3i·18-s + (0.866 + 0.499i)20-s + (−1.99 + 3.46i)22-s + (−2 − 3.46i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−1.30 − 0.755i)7-s + 0.353i·8-s + (0.5 − 0.866i)9-s + (−0.158 − 0.273i)10-s + (1.04 − 0.603i)11-s + 1.06·14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + 0.707i·18-s + (0.193 + 0.111i)20-s + (−0.426 + 0.738i)22-s + (−0.417 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798107 - 0.327593i\)
\(L(\frac12)\) \(\approx\) \(0.798107 - 0.327593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.46 + 2i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.79 + 4.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.92 + 4i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \)
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

−11.26902646039385007318933898395, −10.23064491113738426669289572160, −9.599509808020466987710119194587, −8.823988958816526183095175191946, −7.39209619896648476123467525854, −6.66694449747106026976686636574, −6.06879125257888266636633019758, −4.13502130461993848287457357303, −3.08215926446707170637571756288, −0.792912423538577315458192735290, 1.66040070775267475631262906470, 3.15425685526444171140156879977, 4.49497364421246769370532163952, 5.95943776994714551429821434741, 6.93994845240406028385695019792, 8.051214225885554740725511473465, 9.129861684783321364502906401226, 9.663989473897900537010782482179, 10.54652768999801307921101006696, 11.72421316808822318867774421141

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