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Properties

Label 2-567-21.20-c3-0-66
Degree $2$
Conductor $567$
Sign $0.958 + 0.283i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0979i·2-s + 7.99·4-s + 18.1·5-s + (5.25 − 17.7i)7-s − 1.56i·8-s − 1.77i·10-s + 37.0i·11-s − 18.8i·13-s + (−1.73 − 0.514i)14-s + 63.7·16-s + 62.5·17-s + 70.2i·19-s + 144.·20-s + 3.62·22-s − 162. i·23-s + ⋯
L(s)  = 1  − 0.0346i·2-s + 0.998·4-s + 1.62·5-s + (0.283 − 0.958i)7-s − 0.0691i·8-s − 0.0561i·10-s + 1.01i·11-s − 0.403i·13-s + (−0.0331 − 0.00982i)14-s + 0.996·16-s + 0.891·17-s + 0.848i·19-s + 1.61·20-s + 0.0351·22-s − 1.47i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.958 + 0.283i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ 0.958 + 0.283i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.748324635\)
\(L(\frac12)\) \(\approx\) \(3.748324635\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.25 + 17.7i)T \)
good2 \( 1 + 0.0979iT - 8T^{2} \)
5 \( 1 - 18.1T + 125T^{2} \)
11 \( 1 - 37.0iT - 1.33e3T^{2} \)
13 \( 1 + 18.8iT - 2.19e3T^{2} \)
17 \( 1 - 62.5T + 4.91e3T^{2} \)
19 \( 1 - 70.2iT - 6.85e3T^{2} \)
23 \( 1 + 162. iT - 1.21e4T^{2} \)
29 \( 1 - 95.2iT - 2.43e4T^{2} \)
31 \( 1 - 127. iT - 2.97e4T^{2} \)
37 \( 1 + 378.T + 5.06e4T^{2} \)
41 \( 1 - 198.T + 6.89e4T^{2} \)
43 \( 1 + 321.T + 7.95e4T^{2} \)
47 \( 1 - 158.T + 1.03e5T^{2} \)
53 \( 1 + 191. iT - 1.48e5T^{2} \)
59 \( 1 + 213.T + 2.05e5T^{2} \)
61 \( 1 - 220. iT - 2.26e5T^{2} \)
67 \( 1 + 136.T + 3.00e5T^{2} \)
71 \( 1 - 458. iT - 3.57e5T^{2} \)
73 \( 1 + 967. iT - 3.89e5T^{2} \)
79 \( 1 + 596.T + 4.93e5T^{2} \)
83 \( 1 - 361.T + 5.71e5T^{2} \)
89 \( 1 + 35.4T + 7.04e5T^{2} \)
97 \( 1 - 1.32e3iT - 9.12e5T^{2} \)
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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

−10.28102405523776579129133123267, −9.863731453803901156191055486027, −8.503432677360197398474647015706, −7.37720587048833859205104894578, −6.70940724636105094351968785372, −5.80199489926152687030462880016, −4.85634426195848764264747633993, −3.30273391211493420178621311767, −2.05887465609804190315447035049, −1.29352575662147120607410830007, 1.40424307273911102747937378800, 2.28739932293698299257901434451, 3.23756263021657112202278424833, 5.28779523305425803362765221898, 5.78704304337701898498974639159, 6.50190244013637547743379482000, 7.64735327569866826153011804192, 8.779441466227123802458053378384, 9.518385570665242873056806762129, 10.35255292531526044327180101759

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