Properties

Label 2-38-19.7-c9-0-4
Degree $2$
Conductor $38$
Sign $-0.996 + 0.0781i$
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 + (110. + 191. i)3-s + (−127. + 221. i)4-s + (1.16e3 + 2.00e3i)5-s + (−1.76e3 + 3.06e3i)6-s − 7.71e3·7-s − 4.09e3·8-s + (−1.46e4 + 2.53e4i)9-s + (−1.85e4 + 3.21e4i)10-s + 5.30e4·11-s − 5.66e4·12-s + (6.73e4 − 1.16e5i)13-s + (−6.17e4 − 1.06e5i)14-s + (−2.56e5 + 4.44e5i)15-s + (−3.27e4 − 5.67e4i)16-s + (1.48e5 + 2.57e5i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.788 + 1.36i)3-s + (−0.249 + 0.433i)4-s + (0.830 + 1.43i)5-s + (−0.557 + 0.965i)6-s − 1.21·7-s − 0.353·8-s + (−0.743 + 1.28i)9-s + (−0.587 + 1.01i)10-s + 1.09·11-s − 0.788·12-s + (0.654 − 1.13i)13-s + (−0.429 − 0.743i)14-s + (−1.30 + 2.26i)15-s + (−0.125 − 0.216i)16-s + (0.432 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.996 + 0.0781i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ -0.996 + 0.0781i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.119187 - 3.04482i\)
\(L(\frac12)\) \(\approx\) \(0.119187 - 3.04482i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-5.44e5 + 1.63e5i)T \)
good3 \( 1 + (-110. - 191. i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (-1.16e3 - 2.00e3i)T + (-9.76e5 + 1.69e6i)T^{2} \)
7 \( 1 + 7.71e3T + 4.03e7T^{2} \)
11 \( 1 - 5.30e4T + 2.35e9T^{2} \)
13 \( 1 + (-6.73e4 + 1.16e5i)T + (-5.30e9 - 9.18e9i)T^{2} \)
17 \( 1 + (-1.48e5 - 2.57e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
23 \( 1 + (-1.03e6 + 1.79e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (6.80e5 - 1.17e6i)T + (-7.25e12 - 1.25e13i)T^{2} \)
31 \( 1 + 7.94e6T + 2.64e13T^{2} \)
37 \( 1 + 1.61e7T + 1.29e14T^{2} \)
41 \( 1 + (1.58e5 + 2.75e5i)T + (-1.63e14 + 2.83e14i)T^{2} \)
43 \( 1 + (-1.90e7 - 3.29e7i)T + (-2.51e14 + 4.35e14i)T^{2} \)
47 \( 1 + (-1.64e7 + 2.84e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.95e7 + 3.37e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (-1.98e7 - 3.44e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-1.73e7 + 3.01e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (6.40e7 - 1.10e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-1.00e8 - 1.73e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (8.97e7 + 1.55e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.19e6 - 3.80e6i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + 3.25e8T + 1.86e17T^{2} \)
89 \( 1 + (-1.85e8 + 3.20e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + (3.76e8 + 6.51e8i)T + (-3.80e17 + 6.58e17i)T^{2} \)
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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

−14.78554141427123778560975729144, −14.25593030320304925423314571049, −12.99570124993443731231554996251, −10.75154349287495974848102722811, −9.896263270300512660787485605100, −8.884414151508369881402427653972, −6.91196043919737208667199495754, −5.73891879690666062147560893870, −3.62372303450258205683931938046, −3.01329490038640654737124803567, 0.981853144602188827030834619454, 1.78415191173452650446935879944, 3.54145633464881430197419973603, 5.68568094448518480566449585439, 7.04643069879891568012994882060, 9.132727418530538209646763330302, 9.289921929659568764489055989133, 11.88910427054762626848884401094, 12.65761507285848934814450168242, 13.61056402341584800612524935371

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