Properties

Label 2-19-19.3-c6-0-7
Degree $2$
Conductor $19$
Sign $0.215 + 0.976i$
Analytic cond. $4.37102$
Root an. cond. $2.09070$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.86 − 10.5i)2-s + (10.1 + 27.9i)3-s + (−21.9 − 124. i)4-s + (30.9 − 175. i)5-s + (384. + 140. i)6-s + (131. + 227. i)7-s + (−743. − 429. i)8-s + (−117. + 98.1i)9-s + (−1.57e3 − 1.88e3i)10-s + (−715. + 1.23e3i)11-s + (3.24e3 − 1.87e3i)12-s + (−1.24e3 + 3.42e3i)13-s + (3.57e3 + 629. i)14-s + (5.20e3 − 918. i)15-s + (−3.53e3 + 1.28e3i)16-s + (−2.32e3 − 1.94e3i)17-s + ⋯
L(s)  = 1  + (1.10 − 1.32i)2-s + (0.376 + 1.03i)3-s + (−0.342 − 1.94i)4-s + (0.247 − 1.40i)5-s + (1.78 + 0.648i)6-s + (0.383 + 0.663i)7-s + (−1.45 − 0.838i)8-s + (−0.160 + 0.134i)9-s + (−1.57 − 1.88i)10-s + (−0.537 + 0.931i)11-s + (1.87 − 1.08i)12-s + (−0.568 + 1.56i)13-s + (1.30 + 0.229i)14-s + (1.54 − 0.272i)15-s + (−0.862 + 0.313i)16-s + (−0.472 − 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.215 + 0.976i$
Analytic conductor: \(4.37102\)
Root analytic conductor: \(2.09070\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3),\ 0.215 + 0.976i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.18540 - 1.75651i\)
\(L(\frac12)\) \(\approx\) \(2.18540 - 1.75651i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-5.43e3 - 4.18e3i)T \)
good2 \( 1 + (-8.86 + 10.5i)T + (-11.1 - 63.0i)T^{2} \)
3 \( 1 + (-10.1 - 27.9i)T + (-558. + 468. i)T^{2} \)
5 \( 1 + (-30.9 + 175. i)T + (-1.46e4 - 5.34e3i)T^{2} \)
7 \( 1 + (-131. - 227. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (715. - 1.23e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (1.24e3 - 3.42e3i)T + (-3.69e6 - 3.10e6i)T^{2} \)
17 \( 1 + (2.32e3 + 1.94e3i)T + (4.19e6 + 2.37e7i)T^{2} \)
23 \( 1 + (2.92e3 + 1.65e4i)T + (-1.39e8 + 5.06e7i)T^{2} \)
29 \( 1 + (2.00e4 + 2.39e4i)T + (-1.03e8 + 5.85e8i)T^{2} \)
31 \( 1 + (1.82e4 - 1.05e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 5.60e4iT - 2.56e9T^{2} \)
41 \( 1 + (8.61e3 + 2.36e4i)T + (-3.63e9 + 3.05e9i)T^{2} \)
43 \( 1 + (-3.66e3 + 2.07e4i)T + (-5.94e9 - 2.16e9i)T^{2} \)
47 \( 1 + (-5.15e4 + 4.32e4i)T + (1.87e9 - 1.06e10i)T^{2} \)
53 \( 1 + (5.87e4 - 1.03e4i)T + (2.08e10 - 7.58e9i)T^{2} \)
59 \( 1 + (7.85e4 - 9.36e4i)T + (-7.32e9 - 4.15e10i)T^{2} \)
61 \( 1 + (4.40e4 + 2.49e5i)T + (-4.84e10 + 1.76e10i)T^{2} \)
67 \( 1 + (-1.50e5 - 1.79e5i)T + (-1.57e10 + 8.90e10i)T^{2} \)
71 \( 1 + (2.59e5 + 4.57e4i)T + (1.20e11 + 4.38e10i)T^{2} \)
73 \( 1 + (-1.07e5 + 3.92e4i)T + (1.15e11 - 9.72e10i)T^{2} \)
79 \( 1 + (4.40e4 + 1.21e5i)T + (-1.86e11 + 1.56e11i)T^{2} \)
83 \( 1 + (1.02e5 + 1.78e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (3.07e5 - 8.45e5i)T + (-3.80e11 - 3.19e11i)T^{2} \)
97 \( 1 + (-5.27e5 + 6.28e5i)T + (-1.44e11 - 8.20e11i)T^{2} \)
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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

−16.58170205179781894659283651427, −15.23355282293971169040646836555, −14.07250481211834926625261442317, −12.64978358897001234739235965177, −11.79172561442887753658102509509, −9.988017283334022196802772221273, −9.070790786845560976904888081009, −5.06125098475258469321403594447, −4.32636008966659499381427617870, −2.02097562514041669456306781224, 3.13380551482799606888081905766, 5.68183006717552652650007656099, 7.21873531294799872984979536099, 7.75215138506291447309002295186, 10.83015721382435273362848782511, 12.95099876704469697363610348851, 13.71897890703754756393196412855, 14.59851898553490377875967266348, 15.72297046344358177601928524730, 17.47248565472659111985417060336

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