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Properties

Label 2-38-19.12-c2-0-1
Degree $2$
Conductor $38$
Sign $0.902 + 0.429i$
Analytic cond. $1.03542$
Root an. cond. $1.01755$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.275 − 0.158i)3-s + (0.999 − 1.73i)4-s + (0.5 + 0.866i)5-s + (0.224 − 0.389i)6-s − 2.89·7-s − 2.82i·8-s + (−4.44 + 7.70i)9-s + (1.22 + 0.707i)10-s − 5.10·11-s − 0.635i·12-s + (−0.151 − 0.0874i)13-s + (−3.55 + 2.04i)14-s + (0.275 + 0.158i)15-s + (−2.00 − 3.46i)16-s + (5.94 + 10.3i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0917 − 0.0529i)3-s + (0.249 − 0.433i)4-s + (0.100 + 0.173i)5-s + (0.0374 − 0.0648i)6-s − 0.414·7-s − 0.353i·8-s + (−0.494 + 0.856i)9-s + (0.122 + 0.0707i)10-s − 0.463·11-s − 0.0529i·12-s + (−0.0116 − 0.00672i)13-s + (−0.253 + 0.146i)14-s + (0.0183 + 0.0105i)15-s + (−0.125 − 0.216i)16-s + (0.349 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.902 + 0.429i$
Analytic conductor: \(1.03542\)
Root analytic conductor: \(1.01755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1),\ 0.902 + 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.33655 - 0.301919i\)
\(L(\frac12)\) \(\approx\) \(1.33655 - 0.301919i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
19 \( 1 + (3.34 + 18.7i)T \)
good3 \( 1 + (-0.275 + 0.158i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 2.89T + 49T^{2} \)
11 \( 1 + 5.10T + 121T^{2} \)
13 \( 1 + (0.151 + 0.0874i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-5.94 - 10.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-8.52 + 14.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-38.5 - 22.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 - 21.9iT - 1.36e3T^{2} \)
41 \( 1 + (46.9 - 27.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (18.6 + 32.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (40.7 - 70.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-48.2 - 27.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (29.9 - 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.0 + 65.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-102. - 59.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-65.4 + 37.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (14.6 + 25.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (57.2 - 33.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 30.6T + 6.88e3T^{2} \)
89 \( 1 + (8.84 + 5.10i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (128. - 74.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−15.87120301163524956844802800883, −14.63786388319073422679018376262, −13.56998308049159535977101488184, −12.60444110219311881998074413152, −11.15039369364732328912766994057, −10.12996627750400454547543888662, −8.343665401853907847747523543459, −6.54716793220235018829036095366, −4.91410324378705851944750514110, −2.77921941275094078160393359809, 3.34959860204515262389270935920, 5.35199061766064279139729300468, 6.79350916841984538312736108988, 8.448936186739135465899716450756, 9.919462666143341739660785609278, 11.65107505188385271562360419399, 12.70475478683037092420807500567, 13.89475242032038332520829494831, 14.96844831317446141665052940904, 16.02723857640746481222785174909

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