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Properties

Label 2-567-189.131-c1-0-6
Degree $2$
Conductor $567$
Sign $0.685 + 0.727i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  + (−1.06 − 1.26i)2-s + (−0.129 + 0.734i)4-s + (2.93 + 2.46i)5-s + (2.02 − 1.70i)7-s + (−1.79 + 1.03i)8-s − 6.35i·10-s + (−1.84 − 2.20i)11-s + (1.44 + 3.96i)13-s + (−4.31 − 0.750i)14-s + (4.63 + 1.68i)16-s + 3.18·17-s − 2.26i·19-s + (−2.19 + 1.83i)20-s + (−0.827 + 4.69i)22-s + (0.782 + 2.14i)23-s + ⋯
L(s)  = 1  + (−0.753 − 0.897i)2-s + (−0.0647 + 0.367i)4-s + (1.31 + 1.10i)5-s + (0.764 − 0.644i)7-s + (−0.636 + 0.367i)8-s − 2.01i·10-s + (−0.557 − 0.663i)11-s + (0.399 + 1.09i)13-s + (−1.15 − 0.200i)14-s + (1.15 + 0.422i)16-s + 0.773·17-s − 0.520i·19-s + (−0.490 + 0.411i)20-s + (−0.176 + 0.999i)22-s + (0.163 + 0.448i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.685 + 0.727i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20940 - 0.522216i\)
\(L(\frac12)\) \(\approx\) \(1.20940 - 0.522216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.02 + 1.70i)T \)
good2 \( 1 + (1.06 + 1.26i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-2.93 - 2.46i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (1.84 + 2.20i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.44 - 3.96i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
23 \( 1 + (-0.782 - 2.14i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.356 - 0.979i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-8.54 - 1.50i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.0122 + 0.0211i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.99 - 1.09i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.110 + 0.627i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.86 + 10.5i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.91 + 1.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.2 + 3.74i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.50 - 1.32i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.32 + 6.98i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (3.88 + 2.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.92 - 1.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.68 - 8.13i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-4.16 - 1.51i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 4.73T + 89T^{2} \)
97 \( 1 + (-13.0 + 2.29i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.44025122413132268623765557624, −10.10177736402394979281623975991, −9.149978561048363880879243423851, −8.248218670694485557121827175184, −7.02262338625386740528154526569, −6.13270420041545840036809738291, −5.14499169199773258951613508622, −3.40367459737787582774885658092, −2.35751449009923944208638807382, −1.33568714533659100642811355412, 1.23012924114518262262748225125, 2.70863540796506185619583435423, 4.71856240474814244529226808945, 5.62727346216391243946243426547, 6.12233776972353310333201987570, 7.60106113721545000347690413839, 8.278570443245476542128928588146, 8.869923869257407227367485311570, 9.821821660943956518022014559352, 10.33411870411476510316322681450

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