Properties

Label 2-38-19.11-c9-0-4
Degree $2$
Conductor $38$
Sign $-0.754 - 0.655i$
Analytic cond. $19.5713$
Root an. cond. $4.42395$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 13.8i)2-s + (−43.8 + 75.9i)3-s + (−127. − 221. i)4-s + (993. − 1.72e3i)5-s + (−701. − 1.21e3i)6-s − 8.24e3·7-s + 4.09e3·8-s + (5.99e3 + 1.03e4i)9-s + (1.58e4 + 2.75e4i)10-s + 6.24e4·11-s + 2.24e4·12-s + (−1.25e3 − 2.17e3i)13-s + (6.59e4 − 1.14e5i)14-s + (8.71e4 + 1.50e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (−2.80e5 + 4.86e5i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.312 + 0.541i)3-s + (−0.249 − 0.433i)4-s + (0.711 − 1.23i)5-s + (−0.221 − 0.382i)6-s − 1.29·7-s + 0.353·8-s + (0.304 + 0.527i)9-s + (0.502 + 0.870i)10-s + 1.28·11-s + 0.312·12-s + (−0.0122 − 0.0211i)13-s + (0.458 − 0.794i)14-s + (0.444 + 0.769i)15-s + (−0.125 + 0.216i)16-s + (−0.815 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.754 - 0.655i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ -0.754 - 0.655i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.321554 + 0.860392i\)
\(L(\frac12)\) \(\approx\) \(0.321554 + 0.860392i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 13.8i)T \)
19 \( 1 + (3.40e5 + 4.54e5i)T \)
good3 \( 1 + (43.8 - 75.9i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-993. + 1.72e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + 8.24e3T + 4.03e7T^{2} \)
11 \( 1 - 6.24e4T + 2.35e9T^{2} \)
13 \( 1 + (1.25e3 + 2.17e3i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (2.80e5 - 4.86e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
23 \( 1 + (-7.84e5 - 1.35e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + (-2.89e6 - 5.01e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + 8.75e6T + 2.64e13T^{2} \)
37 \( 1 - 2.54e6T + 1.29e14T^{2} \)
41 \( 1 + (-6.53e6 + 1.13e7i)T + (-1.63e14 - 2.83e14i)T^{2} \)
43 \( 1 + (1.61e7 - 2.79e7i)T + (-2.51e14 - 4.35e14i)T^{2} \)
47 \( 1 + (-2.96e5 - 5.13e5i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (-5.45e7 - 9.44e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (2.22e7 - 3.84e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-5.34e7 - 9.25e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-5.28e7 - 9.15e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + (6.30e7 - 1.09e8i)T + (-2.29e16 - 3.97e16i)T^{2} \)
73 \( 1 + (-1.21e8 + 2.10e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (1.40e8 - 2.43e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 6.00e8T + 1.86e17T^{2} \)
89 \( 1 + (-7.34e7 - 1.27e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + (3.87e8 - 6.71e8i)T + (-3.80e17 - 6.58e17i)T^{2} \)
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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

−15.08149384508862697401886236085, −13.41043153479542799218937485496, −12.71094192386526896399259860090, −10.74500574555722124800402847004, −9.444607100030314910755146678862, −8.871782365524604707625546157124, −6.76995052027596228394573532628, −5.56752838255112986615462694533, −4.18081878707076167731095970701, −1.41595147884850994681773173690, 0.40878182037391628007356011007, 2.23222295175448727304981133062, 3.62676182434036552769729151728, 6.41671393028931453084547132972, 6.86781784596123823333308858073, 9.233787047684407220757575010596, 10.04610556810318623805094821977, 11.38863049924167647690472993513, 12.52747408636343796318388846656, 13.61291732638946899193944167513

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