Properties

Label 2-38-19.18-c2-0-1
Degree $2$
Conductor $38$
Sign $i$
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  − 1.41i·2-s − 2.82i·3-s − 2.00·4-s − 5-s − 4.00·6-s + 5·7-s + 2.82i·8-s + 0.999·9-s + 1.41i·10-s + 5·11-s + 5.65i·12-s + 16.9i·13-s − 7.07i·14-s + 2.82i·15-s + 4.00·16-s − 25·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.942i·3-s − 0.500·4-s − 0.200·5-s − 0.666·6-s + 0.714·7-s + 0.353i·8-s + 0.111·9-s + 0.141i·10-s + 0.454·11-s + 0.471i·12-s + 1.30i·13-s − 0.505i·14-s + 0.188i·15-s + 0.250·16-s − 1.47·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $i$
Analytic conductor: \(1.03542\)
Root analytic conductor: \(1.01755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.735362 - 0.735362i\)
\(L(\frac12)\) \(\approx\) \(0.735362 - 0.735362i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
19 \( 1 - 19T \)
good3 \( 1 + 2.82iT - 9T^{2} \)
5 \( 1 + T + 25T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 - 5T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 + 25T + 289T^{2} \)
23 \( 1 + 10T + 529T^{2} \)
29 \( 1 + 42.4iT - 841T^{2} \)
31 \( 1 - 42.4iT - 961T^{2} \)
37 \( 1 - 25.4iT - 1.36e3T^{2} \)
41 \( 1 - 42.4iT - 1.68e3T^{2} \)
43 \( 1 - 5T + 1.84e3T^{2} \)
47 \( 1 - 5T + 2.20e3T^{2} \)
53 \( 1 - 25.4iT - 2.80e3T^{2} \)
59 \( 1 + 84.8iT - 3.48e3T^{2} \)
61 \( 1 - 95T + 3.72e3T^{2} \)
67 \( 1 + 110. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + 42.4iT - 6.24e3T^{2} \)
83 \( 1 + 130T + 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 16.9iT - 9.40e3T^{2} \)
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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

−15.77567872196033378069628506493, −14.16627309687701457421252395232, −13.39084494492758226668165550448, −11.98348497932857562733541939298, −11.35769899751648216775133019409, −9.568098904237598978724782117449, −8.110799446618265920768383417371, −6.66675494854769332319275468455, −4.39427648108079931839440110889, −1.80246226895369231409882706105, 4.05897326619392014022796515472, 5.44173856762884846101821411921, 7.38922964383083705987277679191, 8.785817704554103179688929946246, 10.08113353934198064649439268782, 11.35388525350450538182918253350, 13.04614192462643548758131361985, 14.43117277388627112367016167710, 15.41560992737907837824552420985, 16.04461245024572773910857014457

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