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Properties

Label 2-567-63.25-c1-0-29
Degree $2$
Conductor $567$
Sign $-0.305 + 0.952i$
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  + 1.27·2-s − 0.386·4-s + (0.776 − 1.34i)5-s + (−1.15 − 2.38i)7-s − 3.03·8-s + (0.986 − 1.70i)10-s + (−1.60 − 2.77i)11-s + (−2.39 − 4.14i)13-s + (−1.46 − 3.02i)14-s − 3.07·16-s + (−1.05 + 1.83i)17-s + (2.43 + 4.21i)19-s + (−0.300 + 0.520i)20-s + (−2.03 − 3.53i)22-s + (1.85 − 3.21i)23-s + ⋯
L(s)  = 1  + 0.898·2-s − 0.193·4-s + (0.347 − 0.601i)5-s + (−0.435 − 0.900i)7-s − 1.07·8-s + (0.311 − 0.540i)10-s + (−0.483 − 0.838i)11-s + (−0.663 − 1.14i)13-s + (−0.391 − 0.808i)14-s − 0.769·16-s + (−0.256 + 0.444i)17-s + (0.557 + 0.966i)19-s + (−0.0671 + 0.116i)20-s + (−0.434 − 0.752i)22-s + (0.386 − 0.669i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897718 - 1.23148i\)
\(L(\frac12)\) \(\approx\) \(0.897718 - 1.23148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 2.38i)T \)
good2 \( 1 - 1.27T + 2T^{2} \)
5 \( 1 + (-0.776 + 1.34i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.60 + 2.77i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.39 + 4.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.05 - 1.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 4.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 3.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.68 + 6.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.50T + 31T^{2} \)
37 \( 1 + (-0.0932 - 0.161i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.39 + 9.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.43 - 4.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.77T + 47T^{2} \)
53 \( 1 + (-0.834 + 1.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.82T + 59T^{2} \)
61 \( 1 - 6.87T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (5.93 - 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.30T + 79T^{2} \)
83 \( 1 + (0.173 - 0.300i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.70 + 15.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.28 + 9.15i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32095337098512118446194818438, −9.831629235459168158831557814330, −8.622109134735080421663319243599, −7.912501987866332852688043066654, −6.56059458151109523012730524748, −5.62525972646710573499241914243, −4.91155056750458468393277822276, −3.81161482455803746970223417114, −2.86665925370199198450550043602, −0.64394607718418973589699909764, 2.38411329521835423141193239240, 3.15497668817837607845249372522, 4.69915735750909827077965750422, 5.17243068284991639889079799527, 6.48773147254283908618995364751, 6.98318818379219687729217849495, 8.517547943085548435251091266787, 9.446853583895258062675822926637, 9.896230638222889061830622423991, 11.28685322480258454313971412251

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