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Properties

Label 2-567-63.38-c1-0-0
Degree $2$
Conductor $567$
Sign $0.617 - 0.786i$
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  − 2.49i·2-s − 4.20·4-s + (−0.617 + 1.07i)5-s + (−1.10 − 2.40i)7-s + 5.49i·8-s + (2.66 + 1.53i)10-s + (−4.75 + 2.74i)11-s + (2.57 − 1.48i)13-s + (−5.99 + 2.74i)14-s + 5.26·16-s + (−2.15 + 3.73i)17-s + (−4.80 + 2.77i)19-s + (2.59 − 4.49i)20-s + (6.83 + 11.8i)22-s + (1.41 + 0.815i)23-s + ⋯
L(s)  = 1  − 1.76i·2-s − 2.10·4-s + (−0.276 + 0.478i)5-s + (−0.416 − 0.909i)7-s + 1.94i·8-s + (0.843 + 0.486i)10-s + (−1.43 + 0.827i)11-s + (0.712 − 0.411i)13-s + (−1.60 + 0.733i)14-s + 1.31·16-s + (−0.523 + 0.906i)17-s + (−1.10 + 0.636i)19-s + (0.580 − 1.00i)20-s + (1.45 + 2.52i)22-s + (0.294 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0616326 + 0.0299816i\)
\(L(\frac12)\) \(\approx\) \(0.0616326 + 0.0299816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 + 2.40i)T \)
good2 \( 1 + 2.49iT - 2T^{2} \)
5 \( 1 + (0.617 - 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.75 - 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.57 + 1.48i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.15 - 3.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.80 - 2.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.41 - 0.815i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.440 + 0.254i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.13iT - 31T^{2} \)
37 \( 1 + (-1.53 - 2.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.177 + 0.306i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0318 + 0.0552i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.86T + 47T^{2} \)
53 \( 1 + (3.57 + 2.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.07T + 59T^{2} \)
61 \( 1 + 6.89iT - 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + (6.09 + 3.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.331T + 79T^{2} \)
83 \( 1 + (3.69 - 6.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.71 + 2.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.85 - 2.22i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.76447881585241187463672018115, −10.35762573282260282599351924948, −9.622798929269049495587515633214, −8.394386426458047783362347681108, −7.56277701606203178030346687296, −6.24340272636662145989210535784, −4.74843670965900273588275001499, −3.87683623494540808507353248993, −2.99291149019441613862208943482, −1.77704648410311550017307190382, 0.03788619948403333798642383304, 2.86704358431968898285181267000, 4.52469216753055550904005716872, 5.24271165080354788659120617363, 6.14343571431347950169825186118, 6.88119451215351649703959010447, 8.013738130803149668307384032904, 8.734374500155593787095087025868, 9.043719679898791934559762428046, 10.45400562759285754361507580656

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