Properties

Label 2-19-19.17-c3-0-3
Degree $2$
Conductor $19$
Sign $0.0208 + 0.999i$
Analytic cond. $1.12103$
Root an. cond. $1.05879$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.477 − 2.70i)2-s + (−4.62 − 3.87i)3-s + (0.412 + 0.150i)4-s + (5.11 − 1.86i)5-s + (−12.7 + 10.6i)6-s + (11.7 + 20.3i)7-s + (11.6 − 20.0i)8-s + (1.63 + 9.27i)9-s + (−2.59 − 14.7i)10-s + (−23.6 + 40.8i)11-s + (−1.32 − 2.29i)12-s + (−14.0 + 11.7i)13-s + (60.6 − 22.0i)14-s + (−30.8 − 11.2i)15-s + (−46.1 − 38.7i)16-s + (16.8 − 95.7i)17-s + ⋯
L(s)  = 1  + (0.168 − 0.957i)2-s + (−0.889 − 0.746i)3-s + (0.0516 + 0.0187i)4-s + (0.457 − 0.166i)5-s + (−0.864 + 0.725i)6-s + (0.633 + 1.09i)7-s + (0.512 − 0.888i)8-s + (0.0605 + 0.343i)9-s + (−0.0821 − 0.465i)10-s + (−0.646 + 1.12i)11-s + (−0.0318 − 0.0552i)12-s + (−0.299 + 0.251i)13-s + (1.15 − 0.421i)14-s + (−0.530 − 0.193i)15-s + (−0.721 − 0.605i)16-s + (0.240 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0208 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0208 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.0208 + 0.999i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ 0.0208 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.757607 - 0.741947i\)
\(L(\frac12)\) \(\approx\) \(0.757607 - 0.741947i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (25.4 - 78.8i)T \)
good2 \( 1 + (-0.477 + 2.70i)T + (-7.51 - 2.73i)T^{2} \)
3 \( 1 + (4.62 + 3.87i)T + (4.68 + 26.5i)T^{2} \)
5 \( 1 + (-5.11 + 1.86i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-11.7 - 20.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (23.6 - 40.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (14.0 - 11.7i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-16.8 + 95.7i)T + (-4.61e3 - 1.68e3i)T^{2} \)
23 \( 1 + (-84.7 - 30.8i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-16.0 - 91.2i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (142. + 246. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 232.T + 5.06e4T^{2} \)
41 \( 1 + (197. + 165. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (75.7 - 27.5i)T + (6.09e4 - 5.11e4i)T^{2} \)
47 \( 1 + (85.6 + 485. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + (106. + 38.8i)T + (1.14e5 + 9.56e4i)T^{2} \)
59 \( 1 + (31.3 - 177. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (-259. - 94.4i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-26.9 - 152. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (577. - 210. i)T + (2.74e5 - 2.30e5i)T^{2} \)
73 \( 1 + (-368. - 309. i)T + (6.75e4 + 3.83e5i)T^{2} \)
79 \( 1 + (-975. - 818. i)T + (8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (154. + 266. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-32.1 + 26.9i)T + (1.22e5 - 6.94e5i)T^{2} \)
97 \( 1 + (-98.6 + 559. i)T + (-8.57e5 - 3.12e5i)T^{2} \)
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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

−18.10429610197730198780188199831, −16.75368433483603873772482623653, −15.09223747983421154214837949996, −13.09514191634871887156374536847, −12.14654856562785850390565959699, −11.40181237116285070934823569994, −9.681027570734621469993570719502, −7.25709112743548963070043192620, −5.36224154073208569630255776232, −2.01401151355066298006743062483, 4.88527830381638644947394285782, 6.21168813744426679540195056024, 7.979217887726098386679367197819, 10.53308043804299441271528904101, 11.05117490820681864201941273315, 13.49479868271364836375908381551, 14.71824807933223314861108169808, 16.00046767615121531612805351352, 16.93356885182216919351569912414, 17.58571049231364178993688467743

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