Properties

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

Origins

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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: $\chi_{38} (35, \cdot )$
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 \( 1 + (1.01e3 + 848. i)T + (1.58e5 + 8.98e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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