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Properties

Label 2-38-19.12-c4-0-1
Degree $2$
Conductor $38$
Sign $0.826 - 0.563i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 − 1.41i)2-s + (−7.91 + 4.57i)3-s + (3.99 − 6.92i)4-s + (19.7 + 34.1i)5-s + (−12.9 + 22.3i)6-s + 91.5·7-s − 22.6i·8-s + (1.27 − 2.20i)9-s + (96.6 + 55.8i)10-s − 154.·11-s + 73.1i·12-s + (−5.46 − 3.15i)13-s + (224. − 129. i)14-s + (−312. − 180. i)15-s + (−32.0 − 55.4i)16-s + (−35.4 − 61.3i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.879 + 0.507i)3-s + (0.249 − 0.433i)4-s + (0.789 + 1.36i)5-s + (−0.359 + 0.621i)6-s + 1.86·7-s − 0.353i·8-s + (0.0157 − 0.0272i)9-s + (0.966 + 0.558i)10-s − 1.27·11-s + 0.507i·12-s + (−0.0323 − 0.0186i)13-s + (1.14 − 0.660i)14-s + (−1.38 − 0.801i)15-s + (−0.125 − 0.216i)16-s + (−0.122 − 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.826 - 0.563i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.826 - 0.563i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.76505 + 0.544229i\)
\(L(\frac12)\) \(\approx\) \(1.76505 + 0.544229i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.44 + 1.41i)T \)
19 \( 1 + (-338. - 124. i)T \)
good3 \( 1 + (7.91 - 4.57i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-19.7 - 34.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 91.5T + 2.40e3T^{2} \)
11 \( 1 + 154.T + 1.46e4T^{2} \)
13 \( 1 + (5.46 + 3.15i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (35.4 + 61.3i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-328. + 569. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (756. + 436. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.35e3iT - 9.23e5T^{2} \)
37 \( 1 - 204. iT - 1.87e6T^{2} \)
41 \( 1 + (1.25e3 - 724. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-694. - 1.20e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-396. + 686. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (427. + 246. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-414. + 239. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.24e3 - 2.15e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.15e3 + 1.82e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-408. + 236. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-656. - 1.13e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.82e3 + 3.94e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 7.24e3T + 4.74e7T^{2} \)
89 \( 1 + (2.75e3 + 1.58e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-4.30e3 + 2.48e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.31238146285749865579021417736, −14.47616649297577773946727466670, −13.50311217503857947965517528254, −11.55225915610679399968122561215, −10.93988558644699360667874542896, −10.15449093472635961400657455818, −7.67090398069527467661371003399, −5.82126585276435595883289696114, −4.84606069714706852981575578076, −2.38242542316848745855334425008, 1.44809421975163800676361785178, 5.15660511129629051560265727517, 5.35505661574279026509251301098, 7.49929000261859629676895448610, 8.820849486953176842900707244745, 10.95476328217226016146614243631, 12.03531329878925290614882079816, 13.03151513883539055974414275961, 14.02934287473349210901476316545, 15.47827231600443351667328578196

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