Properties

Degree $2$
Conductor $38$
Sign $-0.462 + 0.886i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 − 1.28i)2-s + (−6.18 − 2.25i)3-s + (0.694 − 3.93i)4-s + (−1.97 − 11.1i)5-s + (−12.3 + 4.50i)6-s + (4.35 − 7.55i)7-s + (−4.00 − 6.92i)8-s + (12.4 + 10.4i)9-s + (−17.3 − 14.5i)10-s + (25.8 + 44.7i)11-s + (−13.1 + 22.7i)12-s + (26.0 − 9.48i)13-s + (−3.02 − 17.1i)14-s + (−12.9 + 73.5i)15-s + (−15.0 − 5.47i)16-s + (89.9 − 75.4i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−1.18 − 0.433i)3-s + (0.0868 − 0.492i)4-s + (−0.176 − 0.999i)5-s + (−0.841 + 0.306i)6-s + (0.235 − 0.407i)7-s + (−0.176 − 0.306i)8-s + (0.462 + 0.387i)9-s + (−0.549 − 0.461i)10-s + (0.708 + 1.22i)11-s + (−0.316 + 0.548i)12-s + (0.555 − 0.202i)13-s + (−0.0578 − 0.327i)14-s + (−0.223 + 1.26i)15-s + (−0.234 − 0.0855i)16-s + (1.28 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.462 + 0.886i$
Motivic weight: \(3\)
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :3/2),\ -0.462 + 0.886i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.588138 - 0.970520i\)
\(L(\frac12)\) \(\approx\) \(0.588138 - 0.970520i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.53 + 1.28i)T \)
19 \( 1 + (66.4 + 49.4i)T \)
good3 \( 1 + (6.18 + 2.25i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (1.97 + 11.1i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (-4.35 + 7.55i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-25.8 - 44.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-26.0 + 9.48i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (-89.9 + 75.4i)T + (853. - 4.83e3i)T^{2} \)
23 \( 1 + (17.2 - 98.0i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (106. + 89.7i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (-1.55 + 2.68i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 91.5T + 5.06e4T^{2} \)
41 \( 1 + (-234. - 85.5i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (71.8 + 407. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (-468. - 393. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + (105. - 599. i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (438. - 367. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (-99.4 + 563. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-228. - 191. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-43.9 - 249. i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 + (-489. - 178. i)T + (2.98e5 + 2.50e5i)T^{2} \)
79 \( 1 + (212. + 77.4i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-190. + 329. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (949. - 345. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (180. - 151. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−15.50487330578240737996793515861, −13.96447971875660636790469142516, −12.59973615494728121847103013223, −12.06118899837255166217620332092, −10.94220969866396150525286089673, −9.371647516922195649260098174743, −7.28006377916408280979355880423, −5.66818470012601980930414069586, −4.42113063449723259283772279558, −1.06648730716889019959033399643, 3.72296144094416059821015388658, 5.67016626590436642342248672287, 6.47453769687974781322213692978, 8.385606301964299410043395545900, 10.55131587722147586510561581719, 11.31322830653127338489339735464, 12.42256702263843836314772889377, 14.19227724061994086234637634829, 14.96752095941658797430985376490, 16.38683878513558989739792820997

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