↑

Properties

Label 2-338-13.3-c1-0-12
Degree $2$
Conductor $338$
Sign $-0.990 + 0.134i$
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  + (−0.5 − 0.866i)2-s + (−1.17 − 2.04i)3-s + (−0.499 + 0.866i)4-s + 0.890·5-s + (−1.17 + 2.04i)6-s + (2.24 − 3.89i)7-s + 0.999·8-s + (−1.27 + 2.21i)9-s + (−0.445 − 0.770i)10-s + (−1.34 − 2.33i)11-s + 2.35·12-s − 4.49·14-s + (−1.04 − 1.81i)15-s + (−0.5 − 0.866i)16-s + (−1.79 + 3.10i)17-s + 2.55·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.680 − 1.17i)3-s + (−0.249 + 0.433i)4-s + 0.398·5-s + (−0.481 + 0.833i)6-s + (0.849 − 1.47i)7-s + 0.353·8-s + (−0.425 + 0.737i)9-s + (−0.140 − 0.243i)10-s + (−0.405 − 0.702i)11-s + 0.680·12-s − 1.20·14-s + (−0.270 − 0.469i)15-s + (−0.125 − 0.216i)16-s + (−0.434 + 0.752i)17-s + 0.602·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.990 + 0.134i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0591482 - 0.878091i\)
\(L(\frac12)\) \(\approx\) \(0.0591482 - 0.878091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (1.17 + 2.04i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.890T + 5T^{2} \)
7 \( 1 + (-2.24 + 3.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.34 + 2.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.79 - 3.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.46 + 2.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.04 - 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.39T + 31T^{2} \)
37 \( 1 + (-0.753 - 1.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.82 - 3.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0854 - 0.148i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.20T + 47T^{2} \)
53 \( 1 - 6.09T + 53T^{2} \)
59 \( 1 + (1.53 - 2.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.98 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.53 + 9.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.04 + 8.74i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 - 2.93T + 83T^{2} \)
89 \( 1 + (6.09 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.14608105384263008869725176254, −10.54954186944788601594078361497, −9.349711692115021532281311666829, −7.973040614447745151644131493495, −7.46496396168262480722535459860, −6.38670363995462338998216423175, −5.17783492135214138652498778721, −3.75978585929932131137859581796, −1.89489794545941285226224055348, −0.78219426093551626854111152313, 2.27473069989448870757423166664, 4.37976511071894739692784579219, 5.31275359542175198000225715849, 5.73686732832643298276357952780, 7.20538237513373635016374488789, 8.469352217397607789713762047119, 9.257290767625265280439145496308, 10.00889186268459831088216006672, 10.92580676488485769887804745892, 11.73511290869653321773724594595

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