Properties

Label 2-38-19.5-c3-0-4
Degree $2$
Conductor $38$
Sign $0.376 + 0.926i$
Analytic cond. $2.24207$
Root an. cond. $1.49735$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 − 0.684i)2-s + (0.900 − 5.10i)3-s + (3.06 − 2.57i)4-s + (−10.9 − 9.19i)5-s + (−1.80 − 10.2i)6-s + (16.0 + 27.8i)7-s + (4.00 − 6.92i)8-s + (0.0987 + 0.0359i)9-s + (−26.8 − 9.78i)10-s + (16.0 − 27.8i)11-s + (−10.3 − 17.9i)12-s + (10.2 + 58.0i)13-s + (49.2 + 41.3i)14-s + (−56.8 + 47.6i)15-s + (2.77 − 15.7i)16-s + (0.248 − 0.0905i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.173 − 0.982i)3-s + (0.383 − 0.321i)4-s + (−0.980 − 0.822i)5-s + (−0.122 − 0.695i)6-s + (0.867 + 1.50i)7-s + (0.176 − 0.306i)8-s + (0.00365 + 0.00133i)9-s + (−0.850 − 0.309i)10-s + (0.440 − 0.763i)11-s + (−0.249 − 0.432i)12-s + (0.218 + 1.23i)13-s + (0.940 + 0.788i)14-s + (−0.978 + 0.821i)15-s + (0.0434 − 0.246i)16-s + (0.00354 − 0.00129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(2.24207\)
Root analytic conductor: \(1.49735\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47693 - 0.993856i\)
\(L(\frac12)\) \(\approx\) \(1.47693 - 0.993856i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 + 0.684i)T \)
19 \( 1 + (82.8 + 0.155i)T \)
good3 \( 1 + (-0.900 + 5.10i)T + (-25.3 - 9.23i)T^{2} \)
5 \( 1 + (10.9 + 9.19i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-16.0 - 27.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-16.0 + 27.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-10.2 - 58.0i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (-0.248 + 0.0905i)T + (3.76e3 - 3.15e3i)T^{2} \)
23 \( 1 + (125. - 105. i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (-69.8 - 25.4i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (113. + 197. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 112.T + 5.06e4T^{2} \)
41 \( 1 + (33.2 - 188. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (-44.7 - 37.5i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (84.3 + 30.6i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + (-297. + 249. i)T + (2.58e4 - 1.46e5i)T^{2} \)
59 \( 1 + (350. - 127. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (398. - 334. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (61.8 + 22.5i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (-405. - 340. i)T + (6.21e4 + 3.52e5i)T^{2} \)
73 \( 1 + (-66.8 + 378. i)T + (-3.65e5 - 1.33e5i)T^{2} \)
79 \( 1 + (-88.6 + 502. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (679. + 1.17e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-51.9 - 294. i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (-1.58e3 + 575. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−15.44924444609217862618815868238, −14.27851436476140910227938821650, −13.03287422985942049369828408448, −11.92190977668266608295427883601, −11.56167544170961607005077807206, −8.895986956264682158797616615207, −7.906640257136663649607612094430, −6.09097825023946246771513363257, −4.36477120070217981850248428167, −1.84255386599715889719200378875, 3.71536276059409109734819686353, 4.52276510434273810660442664150, 6.95737081341427330960280703861, 8.046804453030935798426308594256, 10.36864392830828432108332175889, 10.89261822954665421097407279374, 12.47103492527464107793453948855, 14.14718848399696113026683230143, 14.89993933429320103111530995479, 15.65609748616283677669026194597

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