↑

Properties

Label 2-38-19.11-c5-0-2
Degree 22
Conductor 3838
Sign −0.278−0.960i-0.278 - 0.960i
Analytic cond. 6.094586.09458
Root an. cond. 2.468722.46872
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (−2.72 + 4.72i)3-s + (−7.99 − 13.8i)4-s + (−41.7 + 72.3i)5-s + (10.9 + 18.8i)6-s − 105.·7-s − 63.9·8-s + (106. + 184. i)9-s + (166. + 289. i)10-s − 245.·11-s + 87.2·12-s + (95.4 + 165. i)13-s + (−211. + 366. i)14-s + (−227. − 394. i)15-s + (−128 + 221. i)16-s + (44.7 − 77.4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.174 + 0.302i)3-s + (−0.249 − 0.433i)4-s + (−0.746 + 1.29i)5-s + (0.123 + 0.214i)6-s − 0.816·7-s − 0.353·8-s + (0.438 + 0.760i)9-s + (0.528 + 0.914i)10-s − 0.611·11-s + 0.174·12-s + (0.156 + 0.271i)13-s + (−0.288 + 0.499i)14-s + (−0.261 − 0.452i)15-s + (−0.125 + 0.216i)16-s + (0.0375 − 0.0650i)17-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=((−0.278−0.960i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=((−0.278−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: −0.278−0.960i-0.278 - 0.960i
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ38(11,⋅)\chi_{38} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), −0.278−0.960i)(2,\ 38,\ (\ :5/2),\ -0.278 - 0.960i)

Particular Values

L(3)L(3) ≈\approx 0.514591+0.685092i0.514591 + 0.685092i
L(12)L(\frac12) ≈\approx 0.514591+0.685092i0.514591 + 0.685092i
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−2+3.46i)T 1 + (-2 + 3.46i)T
19 1+(−109.−1.56e3i)T 1 + (-109. - 1.56e3i)T
good3 1+(2.72−4.72i)T+(−121.5−210.i)T2 1 + (2.72 - 4.72i)T + (-121.5 - 210. i)T^{2}
5 1+(41.7−72.3i)T+(−1.56e3−2.70e3i)T2 1 + (41.7 - 72.3i)T + (-1.56e3 - 2.70e3i)T^{2}
7 1+105.T+1.68e4T2 1 + 105.T + 1.68e4T^{2}
11 1+245.T+1.61e5T2 1 + 245.T + 1.61e5T^{2}
13 1+(−95.4−165.i)T+(−1.85e5+3.21e5i)T2 1 + (-95.4 - 165. i)T + (-1.85e5 + 3.21e5i)T^{2}
17 1+(−44.7+77.4i)T+(−7.09e5−1.22e6i)T2 1 + (-44.7 + 77.4i)T + (-7.09e5 - 1.22e6i)T^{2}
23 1+(1.91e3+3.31e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.91e3 + 3.31e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+(1.05e3+1.82e3i)T+(−1.02e7+1.77e7i)T2 1 + (1.05e3 + 1.82e3i)T + (-1.02e7 + 1.77e7i)T^{2}
31 1−4.69e3T+2.86e7T2 1 - 4.69e3T + 2.86e7T^{2}
37 1−7.99e3T+6.93e7T2 1 - 7.99e3T + 6.93e7T^{2}
41 1+(1.03e4−1.79e4i)T+(−5.79e7−1.00e8i)T2 1 + (1.03e4 - 1.79e4i)T + (-5.79e7 - 1.00e8i)T^{2}
43 1+(−1.10e3+1.91e3i)T+(−7.35e7−1.27e8i)T2 1 + (-1.10e3 + 1.91e3i)T + (-7.35e7 - 1.27e8i)T^{2}
47 1+(−1.00e4−1.74e4i)T+(−1.14e8+1.98e8i)T2 1 + (-1.00e4 - 1.74e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.90e4−3.30e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.90e4 - 3.30e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−2.22e3+3.86e3i)T+(−3.57e8−6.19e8i)T2 1 + (-2.22e3 + 3.86e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−1.34e4−2.33e4i)T+(−4.22e8+7.31e8i)T2 1 + (-1.34e4 - 2.33e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(−2.90e3−5.02e3i)T+(−6.75e8+1.16e9i)T2 1 + (-2.90e3 - 5.02e3i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+(2.52e3−4.37e3i)T+(−9.02e8−1.56e9i)T2 1 + (2.52e3 - 4.37e3i)T + (-9.02e8 - 1.56e9i)T^{2}
73 1+(−1.82e4+3.16e4i)T+(−1.03e9−1.79e9i)T2 1 + (-1.82e4 + 3.16e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−4.51e4+7.81e4i)T+(−1.53e9−2.66e9i)T2 1 + (-4.51e4 + 7.81e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+9.79e4T+3.93e9T2 1 + 9.79e4T + 3.93e9T^{2}
89 1+(2.34e4+4.05e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.34e4 + 4.05e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+(−8.08e3+1.40e4i)T+(−4.29e9−7.43e9i)T2 1 + (-8.08e3 + 1.40e4i)T + (-4.29e9 - 7.43e9i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−15.55392161830492627857536596584, −14.42944090768874541255930374148, −13.20317562364886201786784125890, −11.87520501570984544181712609710, −10.68016355376099793068686353581, −10.00817851231574449733072063070, −7.81912026798950291954271029097, −6.27071276433167413059976697433, −4.22263824758429067801024711672, −2.75566889986901798942090679571, 0.44111127808638808496798053669, 3.81236536584864477489035446405, 5.39321988448181505915729968697, 6.99341878588908051240404940588, 8.340200419048235469688924656322, 9.615236703134677380097957738324, 11.78275971777423213504606478998, 12.72384648293756968365428463521, 13.43243030392477721651837419372, 15.43814787620242112029514372498

Graph of the ZZ-function along the critical line