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Properties

Label 2-567-63.5-c1-0-0
Degree $2$
Conductor $567$
Sign $-0.959 - 0.281i$
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  + 1.41i·2-s + (1.22 + 2.12i)5-s + (−2 + 1.73i)7-s + 2.82i·8-s + (−3 + 1.73i)10-s + (−1.22 − 0.707i)11-s + (−4.5 − 2.59i)13-s + (−2.44 − 2.82i)14-s − 4.00·16-s + (2.44 + 4.24i)17-s + (1.5 + 0.866i)19-s + (1.00 − 1.73i)22-s + (4.89 − 2.82i)23-s + (−0.499 + 0.866i)25-s + (3.67 − 6.36i)26-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.547 + 0.948i)5-s + (−0.755 + 0.654i)7-s + 0.999i·8-s + (−0.948 + 0.547i)10-s + (−0.369 − 0.213i)11-s + (−1.24 − 0.720i)13-s + (−0.654 − 0.755i)14-s − 1.00·16-s + (0.594 + 1.02i)17-s + (0.344 + 0.198i)19-s + (0.213 − 0.369i)22-s + (1.02 − 0.589i)23-s + (−0.0999 + 0.173i)25-s + (0.720 − 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.959 - 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202706 + 1.40939i\)
\(L(\frac12)\) \(\approx\) \(0.202706 + 1.40939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good2 \( 1 - 1.41iT - 2T^{2} \)
5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.89 + 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.67 - 6.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + (2.44 - 1.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + (-3.67 - 6.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9 + 5.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.87579236997909794278554072426, −10.29197356852003399584627168085, −9.349173206035524732257704583974, −8.254821429331406466234891574756, −7.39132469748529532324525595573, −6.59313734040670110805099628121, −5.86047095606619329019985918956, −5.12168906145911644977684315480, −3.14568969937910943497655497594, −2.39132530968618151883623222205, 0.78389156437320115505210551011, 2.18714864742988432289021857089, 3.29962644220448399226633083914, 4.55048386511068449465029453993, 5.49078659000906350213672797347, 6.93907624717557402806000097195, 7.44962761329408933710606503655, 9.207435229237047554588977209722, 9.533471155794528122764329318237, 10.25376434520962843033157063385

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