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Properties

Label 2-1064-19.11-c1-0-29
Degree $2$
Conductor $1064$
Sign $-0.965 - 0.260i$
Analytic cond. $8.49608$
Root an. cond. $2.91480$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 − 2.63i)3-s + (−0.464 + 0.804i)5-s − 7-s + (−3.12 − 5.41i)9-s − 5.85·11-s + (−1.75 − 3.03i)13-s + (1.41 + 2.44i)15-s + (−1.52 + 2.63i)17-s + (0.203 + 4.35i)19-s + (−1.52 + 2.63i)21-s + (−3.41 − 5.91i)23-s + (2.06 + 3.58i)25-s − 9.91·27-s + (−1.19 − 2.06i)29-s − 5.85·31-s + ⋯
L(s)  = 1  + (0.878 − 1.52i)3-s + (−0.207 + 0.359i)5-s − 0.377·7-s + (−1.04 − 1.80i)9-s − 1.76·11-s + (−0.485 − 0.841i)13-s + (0.364 + 0.632i)15-s + (−0.368 + 0.639i)17-s + (0.0467 + 0.998i)19-s + (−0.331 + 0.574i)21-s + (−0.711 − 1.23i)23-s + (0.413 + 0.716i)25-s − 1.90·27-s + (−0.221 − 0.384i)29-s − 1.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(8.49608\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1064} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1064,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8250602423\)
\(L(\frac12)\) \(\approx\) \(0.8250602423\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 + (-0.203 - 4.35i)T \)
good3 \( 1 + (-1.52 + 2.63i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.464 - 0.804i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
13 \( 1 + (1.75 + 3.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.52 - 2.63i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.41 + 5.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.19 + 2.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.85T + 31T^{2} \)
37 \( 1 - 8.70T + 37T^{2} \)
41 \( 1 + (-5.45 + 9.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.908 + 1.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.42 + 7.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.553 - 0.958i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.728 + 1.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.00 + 8.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.42 + 5.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.14 + 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.57 - 4.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.68 - 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 + (-8.71 - 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.53 - 9.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.286193879076428026214219701713, −8.146082169849339221669918898786, −7.916930995907211591823687401992, −7.16994089720408719100963710987, −6.23470868179411711022581433137, −5.39249004804217844580270474641, −3.72519994510261431044164661115, −2.73633286179211290243167390978, −2.08008076046379179109420628002, −0.29709937656506128542290183792, 2.46054518230210501021156294273, 3.09790946488693790916703150702, 4.39578096608790744009489497945, 4.78533349417564350916934832592, 5.79913276921451957802205634276, 7.34333443415025061228104723321, 7.991565237479927839926673127769, 8.977895830821377698140243618541, 9.474189345950642642007711479904, 10.10919438997134256649491880581

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