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Properties

Label 2-38-19.8-c4-0-2
Degree $2$
Conductor $38$
Sign $0.448 - 0.893i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (14.2 + 8.25i)3-s + (3.99 + 6.92i)4-s + (−13.5 + 23.4i)5-s + (−23.3 − 40.4i)6-s − 45.5·7-s − 22.6i·8-s + (95.6 + 165. i)9-s + (66.4 − 38.3i)10-s + 108.·11-s + 132. i·12-s + (38.0 − 21.9i)13-s + (111. + 64.4i)14-s + (−387. + 223. i)15-s + (−32.0 + 55.4i)16-s + (248. − 429. i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (1.58 + 0.916i)3-s + (0.249 + 0.433i)4-s + (−0.542 + 0.939i)5-s + (−0.648 − 1.12i)6-s − 0.929·7-s − 0.353i·8-s + (1.18 + 2.04i)9-s + (0.664 − 0.383i)10-s + 0.900·11-s + 0.916i·12-s + (0.225 − 0.130i)13-s + (0.569 + 0.328i)14-s + (−1.72 + 0.994i)15-s + (−0.125 + 0.216i)16-s + (0.858 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.32325 + 0.816703i\)
\(L(\frac12)\) \(\approx\) \(1.32325 + 0.816703i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 + 1.41i)T \)
19 \( 1 + (-168. - 319. i)T \)
good3 \( 1 + (-14.2 - 8.25i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (13.5 - 23.4i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + 45.5T + 2.40e3T^{2} \)
11 \( 1 - 108.T + 1.46e4T^{2} \)
13 \( 1 + (-38.0 + 21.9i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-248. + 429. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (360. + 623. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-542. + 313. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + 383. iT - 9.23e5T^{2} \)
37 \( 1 - 1.27e3iT - 1.87e6T^{2} \)
41 \( 1 + (2.51e3 + 1.44e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-896. + 1.55e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-59.5 - 103. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-760. + 438. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-733. - 423. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.52e3 - 2.63e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (740. - 427. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-4.84e3 - 2.79e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (2.74e3 - 4.75e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (8.84e3 + 5.10e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 1.20e4T + 4.74e7T^{2} \)
89 \( 1 + (2.61e3 - 1.50e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (7.64e3 + 4.41e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−15.75038399614131251885724394046, −14.63057663458972563387653999321, −13.76397528193023300310323494769, −11.93495294829024782629778782596, −10.29539413388570208459908635249, −9.628226576958080891957125935070, −8.376628051318076902239244896291, −7.08598910689789143978894671283, −3.77734845255600120237072354333, −2.86438405039541212734006100477, 1.30089336873373282112205051474, 3.56507619499014893003048627630, 6.54467877136900682997891003871, 7.86428629992521300944743228052, 8.773968201017733315357981496192, 9.663322726041523731015282708822, 12.10343043090446559275326809466, 13.04648846731864234818074951205, 14.19739723885251740988210399671, 15.34850294726365158390464779521

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