↑

Properties

Label 2-13e2-13.11-c2-0-8
Degree $2$
Conductor $169$
Sign $0.996 - 0.0816i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
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  + (0.609 − 2.27i)2-s + (2.51 + 4.35i)3-s + (−1.33 − 0.770i)4-s + (−1.10 − 1.10i)5-s + (11.4 − 3.06i)6-s + (1.70 + 6.34i)7-s + (4.09 − 4.09i)8-s + (−8.11 + 14.0i)9-s + (−3.19 + 1.84i)10-s + (−1.12 − 0.301i)11-s − 7.74i·12-s + 15.4·14-s + (2.04 − 7.61i)15-s + (−9.89 − 17.1i)16-s + (21.4 + 12.3i)17-s + (27.0 + 27.0i)18-s + ⋯
L(s)  = 1  + (0.304 − 1.13i)2-s + (0.837 + 1.45i)3-s + (−0.333 − 0.192i)4-s + (−0.221 − 0.221i)5-s + (1.90 − 0.510i)6-s + (0.243 + 0.907i)7-s + (0.511 − 0.511i)8-s + (−0.901 + 1.56i)9-s + (−0.319 + 0.184i)10-s + (−0.102 − 0.0273i)11-s − 0.645i·12-s + 1.10·14-s + (0.136 − 0.507i)15-s + (−0.618 − 1.07i)16-s + (1.26 + 0.728i)17-s + (1.50 + 1.50i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.996 - 0.0816i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.996 - 0.0816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31883 + 0.0948311i\)
\(L(\frac12)\) \(\approx\) \(2.31883 + 0.0948311i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.609 + 2.27i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-2.51 - 4.35i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.10 + 1.10i)T + 25iT^{2} \)
7 \( 1 + (-1.70 - 6.34i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (1.12 + 0.301i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-21.4 - 12.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.650 - 0.174i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-1.92 + 1.11i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (9.58 + 16.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (40.7 + 40.7i)T + 961iT^{2} \)
37 \( 1 + (-3.84 - 1.02i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-3.12 + 11.6i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (43.5 + 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.3 - 24.3i)T - 2.20e3iT^{2} \)
53 \( 1 - 65.5T + 2.80e3T^{2} \)
59 \( 1 + (19.1 + 71.6i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (45.4 - 78.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.4 - 91.3i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (65.5 - 17.5i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-51.8 + 51.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 49.1T + 6.24e3T^{2} \)
83 \( 1 + (-70.5 - 70.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (-81.4 - 21.8i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-65.3 + 17.5i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−12.35142516046620828081521244828, −11.51638890808604265021704572004, −10.48778063528445262381746296365, −9.794055560467097408541650992350, −8.826123595821120735252065828575, −7.81953802500352050703531451260, −5.55824486186848262527636902809, −4.31161381030572096529307518277, −3.42844829792819469798178130521, −2.24311175538488775303900171861, 1.50146664273068692397678443736, 3.33571818870031232279602261877, 5.21874891922770802586694023401, 6.60445579896155941253877711789, 7.43570526037322654416840549797, 7.72653607801455728472982887234, 8.964826832601624024921996450286, 10.61200681055612556639434711208, 11.81735108317309783498370288053, 12.98749337611687994651875567659

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