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Properties

Label 2-567-21.5-c1-0-26
Degree $2$
Conductor $567$
Sign $-0.549 + 0.835i$
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  + (0.621 + 0.359i)2-s + (−0.742 − 1.28i)4-s + (0.723 − 1.25i)5-s + (−2.37 − 1.16i)7-s − 2.50i·8-s + (0.900 − 0.519i)10-s + (−1.55 + 0.900i)11-s + 2.18i·13-s + (−1.06 − 1.57i)14-s + (−0.585 + 1.01i)16-s + (−1.95 − 3.38i)17-s + (−3.47 − 2.00i)19-s − 2.14·20-s − 1.29·22-s + (−4.91 − 2.83i)23-s + ⋯
L(s)  = 1  + (0.439 + 0.253i)2-s + (−0.371 − 0.642i)4-s + (0.323 − 0.560i)5-s + (−0.898 − 0.438i)7-s − 0.884i·8-s + (0.284 − 0.164i)10-s + (−0.470 + 0.271i)11-s + 0.604i·13-s + (−0.283 − 0.421i)14-s + (−0.146 + 0.253i)16-s + (−0.473 − 0.820i)17-s + (−0.797 − 0.460i)19-s − 0.480·20-s − 0.275·22-s + (−1.02 − 0.591i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503392 - 0.932963i\)
\(L(\frac12)\) \(\approx\) \(0.503392 - 0.932963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.37 + 1.16i)T \)
good2 \( 1 + (-0.621 - 0.359i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.723 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.55 - 0.900i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.18iT - 13T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.47 + 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.91 + 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.80iT - 29T^{2} \)
31 \( 1 + (-2.45 + 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 7.53T + 43T^{2} \)
47 \( 1 + (-1.16 + 2.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.996 - 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 + 8.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 + 1.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.156 - 0.270i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (-2.42 + 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.21 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.21T + 83T^{2} \)
89 \( 1 + (-5.28 + 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.5iT - 97T^{2} \)
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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

−10.25672428919728056021891562483, −9.549063961633267983055241360750, −8.982648874493535403624159338771, −7.61518600248353096469938546951, −6.52815115172699633327610477327, −5.92455868095815158059525698500, −4.70228331649275253971156324023, −4.10281308446029048732042648487, −2.36542969511845758361227618397, −0.50275241361361757440370918627, 2.38903067846280499853617127477, 3.23398168012915009742107213801, 4.24010865652771907131892091012, 5.60044259526005017076061169696, 6.30617781027721040727266886232, 7.53439974542668676097000255908, 8.456056281355439838404760589669, 9.235142331981894721049538853779, 10.40104580256377183134581993437, 10.89999468264091559544567125726

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