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Properties

Label 2-864-32.13-c1-0-37
Degree $2$
Conductor $864$
Sign $0.0387 + 0.999i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (−0.479 + 1.33i)2-s + (−1.54 − 1.27i)4-s + (−0.855 + 0.354i)5-s + (−3.04 + 3.04i)7-s + (2.43 − 1.43i)8-s + (−0.0615 − 1.30i)10-s + (1.47 + 3.56i)11-s + (−0.524 − 0.217i)13-s + (−2.58 − 5.50i)14-s + (0.748 + 3.92i)16-s − 5.34i·17-s + (−4.09 − 1.69i)19-s + (1.76 + 0.544i)20-s + (−5.44 + 0.256i)22-s + (2.03 + 2.03i)23-s + ⋯
L(s)  = 1  + (−0.338 + 0.940i)2-s + (−0.770 − 0.637i)4-s + (−0.382 + 0.158i)5-s + (−1.14 + 1.14i)7-s + (0.860 − 0.508i)8-s + (−0.0194 − 0.413i)10-s + (0.444 + 1.07i)11-s + (−0.145 − 0.0602i)13-s + (−0.692 − 1.47i)14-s + (0.187 + 0.982i)16-s − 1.29i·17-s + (−0.939 − 0.389i)19-s + (0.395 + 0.121i)20-s + (−1.16 + 0.0546i)22-s + (0.424 + 0.424i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.0387 + 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.0387 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0188421 - 0.0181250i\)
\(L(\frac12)\) \(\approx\) \(0.0188421 - 0.0181250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.479 - 1.33i)T \)
3 \( 1 \)
good5 \( 1 + (0.855 - 0.354i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.04 - 3.04i)T - 7iT^{2} \)
11 \( 1 + (-1.47 - 3.56i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.524 + 0.217i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.34iT - 17T^{2} \)
19 \( 1 + (4.09 + 1.69i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.03 - 2.03i)T + 23iT^{2} \)
29 \( 1 + (-3.31 + 8.00i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (7.84 - 3.25i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.96 + 8.96i)T + 41iT^{2} \)
43 \( 1 + (4.72 + 11.3i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + (-3.31 - 8.00i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.0488 + 0.0202i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.0218 - 0.0526i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-2.67 + 6.45i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (0.889 - 0.889i)T - 71iT^{2} \)
73 \( 1 + (4.73 + 4.73i)T + 73iT^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (-2.42 - 1.00i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.26 - 4.26i)T - 89iT^{2} \)
97 \( 1 + 6.74T + 97T^{2} \)
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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

−9.680431041630672212814819176595, −9.109819858890287026143535075895, −8.342821250992468490812953387072, −7.07812156503026308439133039432, −6.80277278032308928368978846880, −5.69221273529338359887058645282, −4.85178394771090408859388857440, −3.65270890797814480937197596079, −2.24543441556991974813106266791, −0.01491128885084443969444843895, 1.36602781813529973386920566152, 3.13741729315981997109728109209, 3.74011249966928430788730588890, 4.61468017184388664898008132040, 6.19923542774616322843648512899, 6.95769668801381054150107900415, 8.260891303902270390095386098484, 8.601607379787496794173261672514, 9.811364353080096164707712373145, 10.38048581191965136537482601192

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