Properties

Degree $2$
Conductor $38$
Sign $-0.329 + 0.944i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.694 − 3.93i)2-s + (0.893 + 0.749i)3-s + (−15.0 − 5.47i)4-s + (61.2 − 22.2i)5-s + (3.57 − 2.99i)6-s + (−47.2 − 81.8i)7-s + (−32 + 55.4i)8-s + (−41.9 − 237. i)9-s + (−45.2 − 256. i)10-s + (229. − 396. i)11-s + (−9.32 − 16.1i)12-s + (−754. + 632. i)13-s + (−355. + 129. i)14-s + (71.3 + 25.9i)15-s + (196. + 164. i)16-s + (368. − 2.09e3i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.0572 + 0.0480i)3-s + (−0.469 − 0.171i)4-s + (1.09 − 0.398i)5-s + (0.0405 − 0.0339i)6-s + (−0.364 − 0.631i)7-s + (−0.176 + 0.306i)8-s + (−0.172 − 0.979i)9-s + (−0.143 − 0.811i)10-s + (0.570 − 0.988i)11-s + (−0.0186 − 0.0323i)12-s + (−1.23 + 1.03i)13-s + (−0.484 + 0.176i)14-s + (0.0818 + 0.0298i)15-s + (0.191 + 0.160i)16-s + (0.309 − 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.329 + 0.944i$
Motivic weight: \(5\)
Character: $\chi_{38} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :5/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.00761 - 1.41817i\)
\(L(\frac12)\) \(\approx\) \(1.00761 - 1.41817i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.694 + 3.93i)T \)
19 \( 1 + (-1.04e3 - 1.17e3i)T \)
good3 \( 1 + (-0.893 - 0.749i)T + (42.1 + 239. i)T^{2} \)
5 \( 1 + (-61.2 + 22.2i)T + (2.39e3 - 2.00e3i)T^{2} \)
7 \( 1 + (47.2 + 81.8i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-229. + 396. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (754. - 632. i)T + (6.44e4 - 3.65e5i)T^{2} \)
17 \( 1 + (-368. + 2.09e3i)T + (-1.33e6 - 4.85e5i)T^{2} \)
23 \( 1 + (-3.61e3 - 1.31e3i)T + (4.93e6 + 4.13e6i)T^{2} \)
29 \( 1 + (-1.27e3 - 7.22e3i)T + (-1.92e7 + 7.01e6i)T^{2} \)
31 \( 1 + (-588. - 1.01e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.76e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.87e3 - 1.57e3i)T + (2.01e7 + 1.14e8i)T^{2} \)
43 \( 1 + (1.99e3 - 726. i)T + (1.12e8 - 9.44e7i)T^{2} \)
47 \( 1 + (-2.21e3 - 1.25e4i)T + (-2.15e8 + 7.84e7i)T^{2} \)
53 \( 1 + (3.11e4 + 1.13e4i)T + (3.20e8 + 2.68e8i)T^{2} \)
59 \( 1 + (-1.60e3 + 9.09e3i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (9.67e3 + 3.52e3i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (-4.97e3 - 2.81e4i)T + (-1.26e9 + 4.61e8i)T^{2} \)
71 \( 1 + (-6.45e4 + 2.34e4i)T + (1.38e9 - 1.15e9i)T^{2} \)
73 \( 1 + (-2.77e4 - 2.33e4i)T + (3.59e8 + 2.04e9i)T^{2} \)
79 \( 1 + (-1.68e4 - 1.41e4i)T + (5.34e8 + 3.03e9i)T^{2} \)
83 \( 1 + (-5.13e4 - 8.88e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-2.16e4 + 1.81e4i)T + (9.69e8 - 5.49e9i)T^{2} \)
97 \( 1 + (-1.40e4 + 7.95e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.34097248313628009021337538050, −13.84708679309476632734743714554, −12.47659996459418269078469888806, −11.36953591562188630685351146918, −9.622880668552726896029558001545, −9.267542389323949810937227432396, −6.83111237265469686966626168968, −5.16386147025453098677307781139, −3.20815884261805260185868994959, −1.04371725816851312523941175600, 2.44934761378763437776912287314, 5.05979323233986313699346990498, 6.30081529557679980365895166354, 7.76306384338492766840162059023, 9.396205394227943847966473089964, 10.40372232515414716613147424394, 12.43190875403994864558728016393, 13.38670880874027907914395463663, 14.65439762284217426688558259907, 15.36429332738623580353735288469

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