Properties

Label 2-13e2-13.10-c3-0-11
Degree $2$
Conductor $169$
Sign $0.454 - 0.890i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.21 − 1.28i)2-s + (1.84 + 3.19i)3-s + (−0.719 + 1.24i)4-s + 0.561i·5-s + (8.17 + 4.71i)6-s + (15.7 + 9.08i)7-s + 24.1i·8-s + (6.71 − 11.6i)9-s + (0.719 + 1.24i)10-s + (−56.0 + 32.3i)11-s − 5.30·12-s + 46.5·14-s + (−1.79 + 1.03i)15-s + (25.2 + 43.6i)16-s + (−12.7 + 22.1i)17-s − 34.3i·18-s + ⋯
L(s)  = 1  + (0.784 − 0.452i)2-s + (0.354 + 0.614i)3-s + (−0.0899 + 0.155i)4-s + 0.0502i·5-s + (0.556 + 0.321i)6-s + (0.849 + 0.490i)7-s + 1.06i·8-s + (0.248 − 0.430i)9-s + (0.0227 + 0.0393i)10-s + (−1.53 + 0.887i)11-s − 0.127·12-s + 0.888·14-s + (−0.0308 + 0.0178i)15-s + (0.393 + 0.682i)16-s + (−0.182 + 0.315i)17-s − 0.450i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.29616 + 1.40586i\)
\(L(\frac12)\) \(\approx\) \(2.29616 + 1.40586i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.21 + 1.28i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-1.84 - 3.19i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 0.561iT - 125T^{2} \)
7 \( 1 + (-15.7 - 9.08i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (56.0 - 32.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (12.7 - 22.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-93.5 - 53.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36.6 - 63.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (87.9 + 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 113. iT - 2.97e4T^{2} \)
37 \( 1 + (99.4 - 57.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (60.3 - 34.8i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-219. + 379. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 31.9iT - 1.03e5T^{2} \)
53 \( 1 - 2.84T + 1.48e5T^{2} \)
59 \( 1 + (-62.0 - 35.8i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-460. + 797. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (384. - 222. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-469. - 270. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 764. iT - 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 - 603. iT - 5.71e5T^{2} \)
89 \( 1 + (-1.00e3 + 579. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (505. + 291. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−12.52162301796260951941393063467, −11.69387078171378262801868522972, −10.61846215781788547899718276255, −9.572849505979202914516065805282, −8.398371894411551527435413131450, −7.49340358223361480395237106511, −5.45878990431343642718564639227, −4.69771896315871604240149088136, −3.48442546242046188707169647344, −2.21389689532875192500074892144, 1.01523314864079918042109118909, 2.93764608499561134664176727321, 4.74614752038153655446309308091, 5.37554636507102464001271446753, 6.96014549793150789531447342578, 7.70520932639997998908488892006, 8.825611941323425082833147064694, 10.37276053641807040943537998894, 11.08375796370976334782098073199, 12.68003931448388831693124747487

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