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Properties

Label 2-38-19.7-c3-0-0
Degree $2$
Conductor $38$
Sign $0.910 - 0.412i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (2.5 + 4.33i)3-s + (−1.99 + 3.46i)4-s + (6 + 10.3i)5-s + (5 − 8.66i)6-s + 8·7-s + 7.99·8-s + (0.999 − 1.73i)9-s + (12 − 20.7i)10-s + 9·11-s − 20·12-s + (−13 + 22.5i)13-s + (−8 − 13.8i)14-s + (−30.0 + 51.9i)15-s + (−8 − 13.8i)16-s + (−57 − 98.7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.481 + 0.833i)3-s + (−0.249 + 0.433i)4-s + (0.536 + 0.929i)5-s + (0.340 − 0.589i)6-s + 0.431·7-s + 0.353·8-s + (0.0370 − 0.0641i)9-s + (0.379 − 0.657i)10-s + 0.246·11-s − 0.481·12-s + (−0.277 + 0.480i)13-s + (−0.152 − 0.264i)14-s + (−0.516 + 0.894i)15-s + (−0.125 − 0.216i)16-s + (−0.813 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28761 + 0.278289i\)
\(L(\frac12)\) \(\approx\) \(1.28761 + 0.278289i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
19 \( 1 + (66.5 - 49.3i)T \)
good3 \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 8T + 343T^{2} \)
11 \( 1 - 9T + 1.33e3T^{2} \)
13 \( 1 + (13 - 22.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (57 + 98.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-39 + 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-102 + 176. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 98T + 2.97e4T^{2} \)
37 \( 1 + 334T + 5.06e4T^{2} \)
41 \( 1 + (88.5 + 153. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-158 - 273. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-246 + 426. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (339 - 587. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-289.5 - 501. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-176 + 304. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (377.5 - 653. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-72.5 - 125. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-158 - 273. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 567T + 5.71e5T^{2} \)
89 \( 1 + (-57 + 98.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-471.5 - 816. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.79344673072152904997313759790, −14.60973975809949612176928169533, −13.79477012140174774009642549862, −12.00217775923006031960173849819, −10.71420165249823358998415755773, −9.814875581255367554197956981493, −8.706774222914865863429419266714, −6.79763651794006797375943152556, −4.38639626357381357373537488245, −2.61049759785726038636040849688, 1.62139973029177866557976345889, 4.97008215825124048539788859130, 6.67426778657871342886957032830, 8.145497968852400726971723120636, 8.954429043077708115769500661252, 10.62611418324146687729592670656, 12.61974175118189989909971753823, 13.34512977626873638597211088267, 14.52408829033646498089762438057, 15.76774923889544842069056797012

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