Properties

Degree $2$
Conductor $38$
Sign $0.782 + 0.622i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 − 2.57i)2-s + (25.6 + 9.34i)3-s + (2.77 − 15.7i)4-s + (−15.8 − 89.7i)5-s + (102. − 37.3i)6-s + (−22.7 + 39.4i)7-s + (−32.0 − 55.4i)8-s + (386. + 324. i)9-s + (−279. − 234. i)10-s + (203. + 353. i)11-s + (218. − 378. i)12-s + (−395. + 144. i)13-s + (31.6 + 179. i)14-s + (432. − 2.45e3i)15-s + (−240. − 87.5i)16-s + (−646. + 542. i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (1.64 + 0.599i)3-s + (0.0868 − 0.492i)4-s + (−0.283 − 1.60i)5-s + (1.16 − 0.424i)6-s + (−0.175 + 0.304i)7-s + (−0.176 − 0.306i)8-s + (1.58 + 1.33i)9-s + (−0.883 − 0.741i)10-s + (0.507 + 0.879i)11-s + (0.438 − 0.759i)12-s + (−0.649 + 0.236i)13-s + (0.0431 + 0.244i)14-s + (0.496 − 2.81i)15-s + (−0.234 − 0.0855i)16-s + (−0.542 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.782 + 0.622i$
Motivic weight: \(5\)
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.95837 - 1.03412i\)
\(L(\frac12)\) \(\approx\) \(2.95837 - 1.03412i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 + 2.57i)T \)
19 \( 1 + (-1.16e3 + 1.05e3i)T \)
good3 \( 1 + (-25.6 - 9.34i)T + (186. + 156. i)T^{2} \)
5 \( 1 + (15.8 + 89.7i)T + (-2.93e3 + 1.06e3i)T^{2} \)
7 \( 1 + (22.7 - 39.4i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-203. - 353. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (395. - 144. i)T + (2.84e5 - 2.38e5i)T^{2} \)
17 \( 1 + (646. - 542. i)T + (2.46e5 - 1.39e6i)T^{2} \)
23 \( 1 + (809. - 4.59e3i)T + (-6.04e6 - 2.20e6i)T^{2} \)
29 \( 1 + (3.06e3 + 2.56e3i)T + (3.56e6 + 2.01e7i)T^{2} \)
31 \( 1 + (-3.73e3 + 6.46e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.25e4T + 6.93e7T^{2} \)
41 \( 1 + (-5.87e3 - 2.13e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + (-182. - 1.03e3i)T + (-1.38e8 + 5.02e7i)T^{2} \)
47 \( 1 + (4.07e3 + 3.42e3i)T + (3.98e7 + 2.25e8i)T^{2} \)
53 \( 1 + (-879. + 4.98e3i)T + (-3.92e8 - 1.43e8i)T^{2} \)
59 \( 1 + (-5.50e3 + 4.61e3i)T + (1.24e8 - 7.04e8i)T^{2} \)
61 \( 1 + (-8.08e3 + 4.58e4i)T + (-7.93e8 - 2.88e8i)T^{2} \)
67 \( 1 + (3.00e4 + 2.51e4i)T + (2.34e8 + 1.32e9i)T^{2} \)
71 \( 1 + (3.78e3 + 2.14e4i)T + (-1.69e9 + 6.17e8i)T^{2} \)
73 \( 1 + (-1.32e4 - 4.81e3i)T + (1.58e9 + 1.33e9i)T^{2} \)
79 \( 1 + (-8.91e4 - 3.24e4i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (1.22e4 - 2.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-4.79e4 + 1.74e4i)T + (4.27e9 - 3.58e9i)T^{2} \)
97 \( 1 + (-9.98e4 + 8.37e4i)T + (1.49e9 - 8.45e9i)T^{2} \)
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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

−15.22174775560765663521675987262, −13.83814617130456077223286000063, −12.99139188177246156556927116543, −11.85420740417864223087661024502, −9.605967567168667649563052176805, −9.198845920612403161508799060174, −7.75804700667122771700241015713, −4.91219931139745764341310627596, −3.84253919261039188116958020677, −1.93193693961976909973044143900, 2.67610965371333717857931479921, 3.62899021203143665671094908777, 6.64864815901453442852256568750, 7.41745659787821849733425253127, 8.688046236655336372664944910156, 10.40477586011048309079678888626, 12.12620294648911647076121796321, 13.62379934575675829745445989989, 14.31297788539534031326632615382, 14.84225416869721841176549003026

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