Properties

Degree $2$
Conductor $38$
Sign $0.936 + 0.351i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 − 1.28i)2-s + (4.68 + 1.70i)3-s + (0.694 − 3.93i)4-s + (0.408 + 2.31i)5-s + (9.36 − 3.40i)6-s + (−1.42 + 2.46i)7-s + (−4.00 − 6.92i)8-s + (−1.65 − 1.39i)9-s + (3.60 + 3.02i)10-s + (−1.90 − 3.29i)11-s + (9.96 − 17.2i)12-s + (−15.4 + 5.60i)13-s + (0.987 + 5.59i)14-s + (−2.03 + 11.5i)15-s + (−15.0 − 5.47i)16-s + (−57.2 + 48.0i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.901 + 0.328i)3-s + (0.0868 − 0.492i)4-s + (0.0364 + 0.206i)5-s + (0.637 − 0.231i)6-s + (−0.0767 + 0.132i)7-s + (−0.176 − 0.306i)8-s + (−0.0614 − 0.0515i)9-s + (0.113 + 0.0955i)10-s + (−0.0521 − 0.0902i)11-s + (0.239 − 0.415i)12-s + (−0.328 + 0.119i)13-s + (0.0188 + 0.106i)14-s + (−0.0350 + 0.198i)15-s + (−0.234 − 0.0855i)16-s + (−0.817 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.936 + 0.351i$
Motivic weight: \(3\)
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.936 + 0.351i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.95073 - 0.354260i\)
\(L(\frac12)\) \(\approx\) \(1.95073 - 0.354260i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.53 + 1.28i)T \)
19 \( 1 + (41.0 - 71.9i)T \)
good3 \( 1 + (-4.68 - 1.70i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (-0.408 - 2.31i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (1.42 - 2.46i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (1.90 + 3.29i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (15.4 - 5.60i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (57.2 - 48.0i)T + (853. - 4.83e3i)T^{2} \)
23 \( 1 + (-6.08 + 34.5i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-59.8 - 50.2i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (-114. + 198. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 401.T + 5.06e4T^{2} \)
41 \( 1 + (-60.9 - 22.1i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (13.5 + 76.6i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (356. + 299. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + (12.3 - 69.8i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (508. - 426. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (27.4 - 155. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-68.6 - 57.6i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (109. + 621. i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 + (-713. - 259. i)T + (2.98e5 + 2.50e5i)T^{2} \)
79 \( 1 + (-774. - 281. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-66.3 + 114. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (1.32e3 - 482. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (1.01e3 - 848. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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

−15.21943721228048186665297880208, −14.62171797808240163355830981556, −13.49779872599423413627586824829, −12.27848963129216886723206532578, −10.84300012217826396498386960086, −9.570315392816273361830536771748, −8.281839744119667752857544449117, −6.25799077158968199849426715053, −4.20956562096483613172292710688, −2.61519432144845590195005176877, 2.77174712317525807257827391304, 4.81679317681798222354397783719, 6.77649391219166874119762914364, 8.081237038111331736482850339108, 9.255131555464170439267461781635, 11.19283492979620421118010466613, 12.77310162914485384013472827876, 13.60711407218952350227370312494, 14.58927877117959855345007385112, 15.63785214671210234811184352009

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