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Properties

Label 2-38-19.18-c8-0-11
Degree $2$
Conductor $38$
Sign $0.0301 - 0.999i$
Analytic cond. $15.4803$
Root an. cond. $3.93451$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.3i·2-s − 132. i·3-s − 128.·4-s − 12.7·5-s − 1.50e3·6-s − 1.21e3·7-s + 1.44e3i·8-s − 1.10e4·9-s + 144. i·10-s − 8.28e3·11-s + 1.69e4i·12-s − 7.54e3i·13-s + 1.37e4i·14-s + 1.69e3i·15-s + 1.63e4·16-s + 3.80e4·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.63i·3-s − 0.500·4-s − 0.0204·5-s − 1.15·6-s − 0.506·7-s + 0.353i·8-s − 1.68·9-s + 0.0144i·10-s − 0.566·11-s + 0.819i·12-s − 0.264i·13-s + 0.358i·14-s + 0.0334i·15-s + 0.250·16-s + 0.455·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0301 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0301 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.0301 - 0.999i$
Analytic conductor: \(15.4803\)
Root analytic conductor: \(3.93451\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :4),\ 0.0301 - 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.388381 + 0.376828i\)
\(L(\frac12)\) \(\approx\) \(0.388381 + 0.376828i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 11.3iT \)
19 \( 1 + (-1.30e5 - 3.93e3i)T \)
good3 \( 1 + 132. iT - 6.56e3T^{2} \)
5 \( 1 + 12.7T + 3.90e5T^{2} \)
7 \( 1 + 1.21e3T + 5.76e6T^{2} \)
11 \( 1 + 8.28e3T + 2.14e8T^{2} \)
13 \( 1 + 7.54e3iT - 8.15e8T^{2} \)
17 \( 1 - 3.80e4T + 6.97e9T^{2} \)
23 \( 1 + 1.28e5T + 7.83e10T^{2} \)
29 \( 1 - 1.66e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.19e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.58e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.10e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.97e6T + 1.16e13T^{2} \)
47 \( 1 + 6.30e6T + 2.38e13T^{2} \)
53 \( 1 + 1.41e7iT - 6.22e13T^{2} \)
59 \( 1 - 1.66e7iT - 1.46e14T^{2} \)
61 \( 1 + 5.17e5T + 1.91e14T^{2} \)
67 \( 1 + 3.30e6iT - 4.06e14T^{2} \)
71 \( 1 - 1.89e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.16e7T + 8.06e14T^{2} \)
79 \( 1 - 3.09e7iT - 1.51e15T^{2} \)
83 \( 1 + 4.97e7T + 2.25e15T^{2} \)
89 \( 1 + 3.06e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.74e7iT - 7.83e15T^{2} \)
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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

−13.30730117552163639801264270668, −12.50117845223538993743686595208, −11.56537516318152629188167106471, −9.980020658269048847810926730223, −8.314520465563673394842654379853, −7.16225074297569326301687085583, −5.62540188171072705749204291485, −3.11614877516939678541835319812, −1.64846336863703184527712016249, −0.21310570179439178974889741121, 3.35240369451854997875051662168, 4.69919182616036200010226768200, 5.95557189544460840699156604677, 7.920768451214083409064833982587, 9.448992744089649123689494289636, 10.07815711306841046339683992235, 11.59272097153483164631512048537, 13.41731807027412773239515469304, 14.63996821430343486315676182202, 15.64127588739495475554278450173

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