Properties

Label 2-38-19.6-c7-0-1
Degree $2$
Conductor $38$
Sign $-0.545 - 0.837i$
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  + (−6.12 + 5.14i)2-s + (−28.7 − 10.4i)3-s + (11.1 − 63.0i)4-s + (17.7 + 100. i)5-s + (230. − 83.7i)6-s + (521. − 903. i)7-s + (256. + 443. i)8-s + (−958. − 803. i)9-s + (−626. − 525. i)10-s + (3.17e3 + 5.50e3i)11-s + (−979. + 1.69e3i)12-s + (175. − 64.0i)13-s + (1.44e3 + 8.21e3i)14-s + (543. − 3.08e3i)15-s + (−3.84e3 − 1.40e3i)16-s + (−2.85e4 + 2.39e4i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.614 − 0.223i)3-s + (0.0868 − 0.492i)4-s + (0.0635 + 0.360i)5-s + (0.434 − 0.158i)6-s + (0.574 − 0.995i)7-s + (0.176 + 0.306i)8-s + (−0.438 − 0.367i)9-s + (−0.198 − 0.166i)10-s + (0.720 + 1.24i)11-s + (−0.163 + 0.283i)12-s + (0.0222 − 0.00808i)13-s + (0.141 + 0.800i)14-s + (0.0415 − 0.235i)15-s + (−0.234 − 0.0855i)16-s + (−1.40 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.545 - 0.837i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -0.545 - 0.837i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.304394 + 0.561556i\)
\(L(\frac12)\) \(\approx\) \(0.304394 + 0.561556i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.12 - 5.14i)T \)
19 \( 1 + (2.73e4 - 1.21e4i)T \)
good3 \( 1 + (28.7 + 10.4i)T + (1.67e3 + 1.40e3i)T^{2} \)
5 \( 1 + (-17.7 - 100. i)T + (-7.34e4 + 2.67e4i)T^{2} \)
7 \( 1 + (-521. + 903. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-3.17e3 - 5.50e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-175. + 64.0i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (2.85e4 - 2.39e4i)T + (7.12e7 - 4.04e8i)T^{2} \)
23 \( 1 + (9.09e3 - 5.16e4i)T + (-3.19e9 - 1.16e9i)T^{2} \)
29 \( 1 + (-2.20e4 - 1.85e4i)T + (2.99e9 + 1.69e10i)T^{2} \)
31 \( 1 + (1.00e5 - 1.74e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 1.43e5T + 9.49e10T^{2} \)
41 \( 1 + (-6.02e5 - 2.19e5i)T + (1.49e11 + 1.25e11i)T^{2} \)
43 \( 1 + (1.56e4 + 8.85e4i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (1.01e6 + 8.48e5i)T + (8.79e10 + 4.98e11i)T^{2} \)
53 \( 1 + (2.74e5 - 1.55e6i)T + (-1.10e12 - 4.01e11i)T^{2} \)
59 \( 1 + (9.80e5 - 8.22e5i)T + (4.32e11 - 2.45e12i)T^{2} \)
61 \( 1 + (1.28e5 - 7.30e5i)T + (-2.95e12 - 1.07e12i)T^{2} \)
67 \( 1 + (-2.37e6 - 1.98e6i)T + (1.05e12 + 5.96e12i)T^{2} \)
71 \( 1 + (8.91e3 + 5.05e4i)T + (-8.54e12 + 3.11e12i)T^{2} \)
73 \( 1 + (4.03e6 + 1.46e6i)T + (8.46e12 + 7.10e12i)T^{2} \)
79 \( 1 + (4.50e6 + 1.64e6i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (-3.45e5 + 5.97e5i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (1.48e6 - 5.42e5i)T + (3.38e13 - 2.84e13i)T^{2} \)
97 \( 1 + (-1.45e6 + 1.22e6i)T + (1.40e13 - 7.95e13i)T^{2} \)
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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

−15.08357025458904650326184929522, −14.35833247475408605694476716573, −12.71743273040601939933737618982, −11.26384557736653959616650468713, −10.36974532493303244907378052827, −8.790977386614971711675631588593, −7.21354995352387262873663220830, −6.28615009234985516130779198444, −4.37348644960252562501947751024, −1.50953817360231837195125998314, 0.37144035178805342305732800693, 2.43647743851174230413281286213, 4.74746094816580602379376999798, 6.24520250561046901905386625410, 8.405364954420114899907409268240, 9.156812050193192389831257102661, 11.14090643039179151037830098646, 11.40281249365787020484860614083, 12.89756313582596064322965830037, 14.36705320483683677646099797081

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