Properties

Label 2-19-19.6-c3-0-0
Degree $2$
Conductor $19$
Sign $-0.996 + 0.0835i$
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  + (−2.26 + 1.89i)2-s + (−8.82 − 3.21i)3-s + (0.124 − 0.707i)4-s + (−0.0925 − 0.524i)5-s + (26.0 − 9.48i)6-s + (−11.8 + 20.6i)7-s + (−10.7 − 18.6i)8-s + (46.9 + 39.3i)9-s + (1.20 + 1.01i)10-s + (−18.5 − 32.1i)11-s + (−3.37 + 5.84i)12-s + (−14.9 + 5.45i)13-s + (−12.1 − 69.1i)14-s + (−0.869 + 4.93i)15-s + (65.0 + 23.6i)16-s + (−47.7 + 40.0i)17-s + ⋯
L(s)  = 1  + (−0.799 + 0.671i)2-s + (−1.69 − 0.618i)3-s + (0.0155 − 0.0884i)4-s + (−0.00827 − 0.0469i)5-s + (1.77 − 0.645i)6-s + (−0.642 + 1.11i)7-s + (−0.475 − 0.822i)8-s + (1.73 + 1.45i)9-s + (0.0381 + 0.0319i)10-s + (−0.509 − 0.882i)11-s + (−0.0812 + 0.140i)12-s + (−0.319 + 0.116i)13-s + (−0.232 − 1.32i)14-s + (−0.0149 + 0.0848i)15-s + (1.01 + 0.369i)16-s + (−0.680 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0835i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.996 + 0.0835i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ -0.996 + 0.0835i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00504702 - 0.120655i\)
\(L(\frac12)\) \(\approx\) \(0.00504702 - 0.120655i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (57.8 - 59.2i)T \)
good2 \( 1 + (2.26 - 1.89i)T + (1.38 - 7.87i)T^{2} \)
3 \( 1 + (8.82 + 3.21i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (0.0925 + 0.524i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (11.8 - 20.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (18.5 + 32.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (14.9 - 5.45i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (47.7 - 40.0i)T + (853. - 4.83e3i)T^{2} \)
23 \( 1 + (3.18 - 18.0i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-13.6 - 11.4i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (76.5 - 132. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 41.5T + 5.06e4T^{2} \)
41 \( 1 + (314. + 114. i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (53.2 + 302. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (-37.2 - 31.2i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + (84.6 - 479. i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (-160. + 134. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (36.3 - 206. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-327. - 274. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-105. - 599. i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 + (299. + 108. i)T + (2.98e5 + 2.50e5i)T^{2} \)
79 \( 1 + (245. + 89.2i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-328. + 569. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (1.24e3 - 453. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (-245. + 206. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−18.49836212930979654182622307496, −17.34449951067221629020400545822, −16.46876483109795519269030473839, −15.60434269083780497356788312558, −12.89944074035852787371913147322, −12.11620547045204563056614855016, −10.50774860748358818220755699033, −8.598247531601124833777252326887, −6.79863374639014299483093880549, −5.73583334947869435320502790474, 0.20860851505968663132450217656, 4.84129615331434657141865323334, 6.78326650046074924374574843558, 9.656771321303331740556373670133, 10.47908194312648436096477953346, 11.37094169929685193531061925485, 12.86979589882644939831673090335, 15.20590431164478513249739626974, 16.61268111558212563967041994272, 17.44575515367664446920581558775

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