Properties

Label 2-38-19.10-c2-0-0
Degree $2$
Conductor $38$
Sign $0.918 - 0.395i$
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  + (−1.39 + 0.245i)2-s + (0.272 + 0.324i)3-s + (1.87 − 0.684i)4-s + (7.98 + 2.90i)5-s + (−0.459 − 0.385i)6-s + (−2.58 + 4.47i)7-s + (−2.44 + 1.41i)8-s + (1.53 − 8.68i)9-s + (−11.8 − 2.08i)10-s + (−4.81 − 8.34i)11-s + (0.734 + 0.423i)12-s + (−8.78 + 10.4i)13-s + (2.50 − 6.87i)14-s + (1.23 + 3.38i)15-s + (3.06 − 2.57i)16-s + (−3.08 − 17.4i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (0.0908 + 0.108i)3-s + (0.469 − 0.171i)4-s + (1.59 + 0.581i)5-s + (−0.0765 − 0.0642i)6-s + (−0.369 + 0.639i)7-s + (−0.306 + 0.176i)8-s + (0.170 − 0.965i)9-s + (−1.18 − 0.208i)10-s + (−0.438 − 0.758i)11-s + (0.0611 + 0.0353i)12-s + (−0.675 + 0.804i)13-s + (0.178 − 0.490i)14-s + (0.0821 + 0.225i)15-s + (0.191 − 0.160i)16-s + (−0.181 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(1.03542\)
Root analytic conductor: \(1.01755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :1),\ 0.918 - 0.395i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.907304 + 0.187172i\)
\(L(\frac12)\) \(\approx\) \(0.907304 + 0.187172i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.245i)T \)
19 \( 1 + (17.6 - 7.11i)T \)
good3 \( 1 + (-0.272 - 0.324i)T + (-1.56 + 8.86i)T^{2} \)
5 \( 1 + (-7.98 - 2.90i)T + (19.1 + 16.0i)T^{2} \)
7 \( 1 + (2.58 - 4.47i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.81 + 8.34i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (8.78 - 10.4i)T + (-29.3 - 166. i)T^{2} \)
17 \( 1 + (3.08 + 17.4i)T + (-271. + 98.8i)T^{2} \)
23 \( 1 + (-14.1 + 5.13i)T + (405. - 340. i)T^{2} \)
29 \( 1 + (-11.5 - 2.04i)T + (790. + 287. i)T^{2} \)
31 \( 1 + (44.1 + 25.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 17.8iT - 1.36e3T^{2} \)
41 \( 1 + (15.8 + 18.8i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-14.2 - 5.18i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (13.4 - 76.3i)T + (-2.07e3 - 755. i)T^{2} \)
53 \( 1 + (-21.5 - 59.1i)T + (-2.15e3 + 1.80e3i)T^{2} \)
59 \( 1 + (-52.2 + 9.21i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-46.5 + 16.9i)T + (2.85e3 - 2.39e3i)T^{2} \)
67 \( 1 + (-9.95 - 1.75i)T + (4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (-38.9 + 107. i)T + (-3.86e3 - 3.24e3i)T^{2} \)
73 \( 1 + (85.5 - 71.7i)T + (925. - 5.24e3i)T^{2} \)
79 \( 1 + (-97.8 - 116. i)T + (-1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-26.5 + 45.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (71.6 - 85.3i)T + (-1.37e3 - 7.80e3i)T^{2} \)
97 \( 1 + (84.8 - 14.9i)T + (8.84e3 - 3.21e3i)T^{2} \)
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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

−16.36643750556758845925861558104, −14.98687659207734432906452512984, −14.02279044349908042309901736860, −12.59369174348003254842944294640, −10.96980073726490423073080165738, −9.651063901258934228887684442320, −9.056993310601006816645090658521, −6.84563418616671926150996731131, −5.79418505964378351407114182592, −2.53563924226137424975832093633, 2.05550907107791831233834312198, 5.21822468332040402531824990671, 6.97396124835732115177278382611, 8.546020804582812917065091809552, 10.00853723530267422737367624548, 10.50404991882106389451153842078, 12.82711126208857331856525584475, 13.26970723727520970724979109177, 14.87460929353527009964084657764, 16.49393207281071775779046421273

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