Properties

Label 2-19-19.13-c2-0-0
Degree $2$
Conductor $19$
Sign $0.896 - 0.442i$
Analytic cond. $0.517712$
Root an. cond. $0.719522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.468 + 0.558i)2-s + (−1.14 + 3.13i)3-s + (0.602 − 3.41i)4-s + (−1.13 − 6.41i)5-s + (−2.28 + 0.832i)6-s + (−5.47 + 9.49i)7-s + (4.71 − 2.72i)8-s + (−1.64 − 1.38i)9-s + (3.05 − 3.63i)10-s + (−2.46 − 4.26i)11-s + (10.0 + 5.79i)12-s + (2.41 + 6.63i)13-s + (−7.86 + 1.38i)14-s + (21.4 + 3.77i)15-s + (−9.31 − 3.38i)16-s + (−4.19 + 3.52i)17-s + ⋯
L(s)  = 1  + (0.234 + 0.279i)2-s + (−0.380 + 1.04i)3-s + (0.150 − 0.854i)4-s + (−0.226 − 1.28i)5-s + (−0.381 + 0.138i)6-s + (−0.782 + 1.35i)7-s + (0.589 − 0.340i)8-s + (−0.183 − 0.153i)9-s + (0.305 − 0.363i)10-s + (−0.223 − 0.387i)11-s + (0.835 + 0.482i)12-s + (0.185 + 0.510i)13-s + (−0.561 + 0.0990i)14-s + (1.42 + 0.251i)15-s + (−0.581 − 0.211i)16-s + (−0.246 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(0.517712\)
Root analytic conductor: \(0.719522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1),\ 0.896 - 0.442i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.820558 + 0.191287i\)
\(L(\frac12)\) \(\approx\) \(0.820558 + 0.191287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-18.4 - 4.50i)T \)
good2 \( 1 + (-0.468 - 0.558i)T + (-0.694 + 3.93i)T^{2} \)
3 \( 1 + (1.14 - 3.13i)T + (-6.89 - 5.78i)T^{2} \)
5 \( 1 + (1.13 + 6.41i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (5.47 - 9.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.46 + 4.26i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.41 - 6.63i)T + (-129. + 108. i)T^{2} \)
17 \( 1 + (4.19 - 3.52i)T + (50.1 - 284. i)T^{2} \)
23 \( 1 + (-3.59 + 20.4i)T + (-497. - 180. i)T^{2} \)
29 \( 1 + (-16.3 + 19.5i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (4.26 + 2.46i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 25.2iT - 1.36e3T^{2} \)
41 \( 1 + (24.5 - 67.3i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-4.59 - 26.0i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (35.0 + 29.4i)T + (383. + 2.17e3i)T^{2} \)
53 \( 1 + (-45.4 - 8.00i)T + (2.63e3 + 960. i)T^{2} \)
59 \( 1 + (-32.2 - 38.3i)T + (-604. + 3.42e3i)T^{2} \)
61 \( 1 + (-0.333 + 1.89i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-2.03 + 2.42i)T + (-779. - 4.42e3i)T^{2} \)
71 \( 1 + (87.4 - 15.4i)T + (4.73e3 - 1.72e3i)T^{2} \)
73 \( 1 + (72.2 + 26.3i)T + (4.08e3 + 3.42e3i)T^{2} \)
79 \( 1 + (-1.72 + 4.75i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-12.7 + 22.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (11.0 + 30.3i)T + (-6.06e3 + 5.09e3i)T^{2} \)
97 \( 1 + (-114. - 135. i)T + (-1.63e3 + 9.26e3i)T^{2} \)
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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

−18.56538774934111624463231737259, −16.30287777769461351275666831445, −16.13817322157359537283017173957, −15.00860133594647693278084903932, −13.15194654034538199838672164504, −11.66728415091192684659175514932, −9.968786190995362723830802723702, −8.891468954006093908700494529256, −5.91534662688780431176772607572, −4.75427582003012185626047363568, 3.37701672274605221597132550458, 6.92625471020993493116535833028, 7.41168908165541680535077079813, 10.38332444176156068625174709695, 11.64680991875190023790865506557, 12.97761599115639775976471137007, 13.82696239851165335162138053203, 15.79130731212135021007000407805, 17.30682460696923896331365320729, 18.15047755514719284244230951231

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