Properties

Label 2-38-19.11-c7-0-0
Degree $2$
Conductor $38$
Sign $-0.0106 + 0.999i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−20.5 + 35.5i)3-s + (−31.9 − 55.4i)4-s + (−273. + 473. i)5-s + (−164. − 284. i)6-s − 154.·7-s + 511.·8-s + (252. + 437. i)9-s + (−2.18e3 − 3.78e3i)10-s − 4.15e3·11-s + 2.62e3·12-s + (4.20e3 + 7.27e3i)13-s + (616. − 1.06e3i)14-s + (−1.12e4 − 1.94e4i)15-s + (−2.04e3 + 3.54e3i)16-s + (1.61e4 − 2.80e4i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.438 + 0.759i)3-s + (−0.249 − 0.433i)4-s + (−0.977 + 1.69i)5-s + (−0.310 − 0.537i)6-s − 0.169·7-s + 0.353·8-s + (0.115 + 0.199i)9-s + (−0.690 − 1.19i)10-s − 0.940·11-s + 0.438·12-s + (0.530 + 0.918i)13-s + (0.0600 − 0.104i)14-s + (−0.856 − 1.48i)15-s + (−0.125 + 0.216i)16-s + (0.798 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.0106 + 0.999i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -0.0106 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.293354 - 0.296509i\)
\(L(\frac12)\) \(\approx\) \(0.293354 - 0.296509i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
19 \( 1 + (-6.62e3 + 2.91e4i)T \)
good3 \( 1 + (20.5 - 35.5i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (273. - 473. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + 154.T + 8.23e5T^{2} \)
11 \( 1 + 4.15e3T + 1.94e7T^{2} \)
13 \( 1 + (-4.20e3 - 7.27e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-1.61e4 + 2.80e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
23 \( 1 + (-3.21e4 - 5.57e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-1.01e5 - 1.75e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + 1.32e5T + 2.75e10T^{2} \)
37 \( 1 - 9.36e4T + 9.49e10T^{2} \)
41 \( 1 + (2.48e5 - 4.30e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (1.72e5 - 2.98e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (2.08e4 + 3.60e4i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-1.64e5 - 2.85e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-8.15e5 + 1.41e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.30e4 + 2.25e4i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.61e6 + 2.79e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-8.19e5 + 1.41e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-9.38e5 + 1.62e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (3.04e5 - 5.27e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 2.79e6T + 2.71e13T^{2} \)
89 \( 1 + (-1.68e6 - 2.91e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (3.68e6 - 6.37e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−15.84125698781334724035714693790, −14.86170993544183078554366417725, −13.70592227553160279656705632244, −11.46177643281873475201792156840, −10.80298372404034362750751454443, −9.634688862532399220096637916158, −7.69295112291989052202526910501, −6.78944813324888340090248559914, −4.95739285956980292222508001551, −3.19691728509211993109018991950, 0.24875947994334817248249856655, 1.26015342867424644584118144123, 3.85625829235374975024061116198, 5.58338245181872808106975035976, 7.79545066384890594327511288712, 8.512003346470387431772770387792, 10.22936904758916399372663722435, 11.78694849640990146220256818356, 12.68312072035705656272814410706, 13.01455125602729628987049424615

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