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Properties

Label 2-864-96.35-c1-0-46
Degree $2$
Conductor $864$
Sign $-0.201 + 0.979i$
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  + (−0.919 + 1.07i)2-s + (−0.307 − 1.97i)4-s + (−1.92 + 0.796i)5-s + (0.893 + 0.893i)7-s + (2.40 + 1.48i)8-s + (0.912 − 2.79i)10-s + (−2.30 + 0.956i)11-s + (1.00 − 2.43i)13-s + (−1.78 + 0.138i)14-s + (−3.81 + 1.21i)16-s − 0.737·17-s + (−0.939 − 0.389i)19-s + (2.16 + 3.55i)20-s + (1.09 − 3.35i)22-s + (−2.03 − 2.03i)23-s + ⋯
L(s)  = 1  + (−0.650 + 0.759i)2-s + (−0.153 − 0.988i)4-s + (−0.859 + 0.356i)5-s + (0.337 + 0.337i)7-s + (0.850 + 0.525i)8-s + (0.288 − 0.884i)10-s + (−0.696 + 0.288i)11-s + (0.279 − 0.674i)13-s + (−0.476 + 0.0368i)14-s + (−0.952 + 0.304i)16-s − 0.178·17-s + (−0.215 − 0.0893i)19-s + (0.484 + 0.794i)20-s + (0.233 − 0.716i)22-s + (−0.425 − 0.425i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105248 - 0.129059i\)
\(L(\frac12)\) \(\approx\) \(0.105248 - 0.129059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.919 - 1.07i)T \)
3 \( 1 \)
good5 \( 1 + (1.92 - 0.796i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.893 - 0.893i)T + 7iT^{2} \)
11 \( 1 + (2.30 - 0.956i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.00 + 2.43i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.737T + 17T^{2} \)
19 \( 1 + (0.939 + 0.389i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.03 + 2.03i)T + 23iT^{2} \)
29 \( 1 + (1.71 - 4.13i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.70iT - 31T^{2} \)
37 \( 1 + (2.62 + 6.33i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.488 - 0.488i)T - 41iT^{2} \)
43 \( 1 + (2.31 + 5.58i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.29iT - 47T^{2} \)
53 \( 1 + (4.28 + 10.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.69 + 11.3i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (9.31 + 3.85i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.83 - 11.6i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.02 + 7.02i)T - 71iT^{2} \)
73 \( 1 + (2.71 + 2.71i)T + 73iT^{2} \)
79 \( 1 - 1.95T + 79T^{2} \)
83 \( 1 + (3.06 - 7.39i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.22 + 5.22i)T + 89iT^{2} \)
97 \( 1 + 4.63T + 97T^{2} \)
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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

−9.900799327650387244846872767901, −8.815377426176804617208709887745, −8.164528308288906824841188434941, −7.49889851302558246277487836056, −6.70199393903712710875879571309, −5.59365862909716257822490698529, −4.82557327469811426149466723683, −3.52902831858215423325630647113, −2.00808684651439252114937895506, −0.10194197148821621000810706417, 1.46381088504369315632176150629, 2.86785254664770136977618828058, 4.04062375978578621811029594432, 4.63875498927857476517461440697, 6.17096358705085426156691563852, 7.48255218450303502155946700986, 7.938992084063485929708090180341, 8.717009466676440687087400523893, 9.579145587407576310092014300495, 10.50512280421256410277854780247

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