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Properties

Label 2-864-96.11-c1-0-30
Degree $2$
Conductor $864$
Sign $0.995 - 0.0904i$
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  + (0.0811 + 1.41i)2-s + (−1.98 + 0.229i)4-s + (−1.02 − 0.423i)5-s + (−0.549 + 0.549i)7-s + (−0.484 − 2.78i)8-s + (0.515 − 1.47i)10-s + (2.44 + 1.01i)11-s + (−2.24 − 5.41i)13-s + (−0.820 − 0.731i)14-s + (3.89 − 0.910i)16-s − 0.515·17-s + (2.66 − 1.10i)19-s + (2.12 + 0.607i)20-s + (−1.23 + 3.53i)22-s + (3.26 − 3.26i)23-s + ⋯
L(s)  = 1  + (0.0573 + 0.998i)2-s + (−0.993 + 0.114i)4-s + (−0.457 − 0.189i)5-s + (−0.207 + 0.207i)7-s + (−0.171 − 0.985i)8-s + (0.162 − 0.467i)10-s + (0.738 + 0.305i)11-s + (−0.622 − 1.50i)13-s + (−0.219 − 0.195i)14-s + (0.973 − 0.227i)16-s − 0.124·17-s + (0.611 − 0.253i)19-s + (0.476 + 0.135i)20-s + (−0.262 + 0.754i)22-s + (0.681 − 0.681i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.995 - 0.0904i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.995 - 0.0904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13155 + 0.0512669i\)
\(L(\frac12)\) \(\approx\) \(1.13155 + 0.0512669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0811 - 1.41i)T \)
3 \( 1 \)
good5 \( 1 + (1.02 + 0.423i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.549 - 0.549i)T - 7iT^{2} \)
11 \( 1 + (-2.44 - 1.01i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.24 + 5.41i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.515T + 17T^{2} \)
19 \( 1 + (-2.66 + 1.10i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.26 + 3.26i)T - 23iT^{2} \)
29 \( 1 + (-2.83 - 6.84i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 5.60iT - 31T^{2} \)
37 \( 1 + (-2.58 + 6.24i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-8.45 - 8.45i)T + 41iT^{2} \)
43 \( 1 + (-1.31 + 3.17i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.34iT - 47T^{2} \)
53 \( 1 + (-3.08 + 7.44i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.96 + 7.14i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.36 + 2.22i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.398 - 0.962i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.43 + 5.43i)T + 71iT^{2} \)
73 \( 1 + (-2.98 + 2.98i)T - 73iT^{2} \)
79 \( 1 + 6.13T + 79T^{2} \)
83 \( 1 + (1.05 + 2.55i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-0.656 + 0.656i)T - 89iT^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−9.875008771415580286203209632602, −9.234458509694355268673685452083, −8.298138438849331778184127146014, −7.61940970745007893173515434426, −6.81060691395357373025440004740, −5.83703802919689180352504035055, −4.97652893804129446037618696953, −4.05774430671439243905865136612, −2.89323294269337255505009556237, −0.63780501934014202261091138005, 1.25102800951863598233456040011, 2.62162976466443053013649278329, 3.77832487216857651307199083640, 4.39845232918710471013093628794, 5.59546361762350683045274044819, 6.78321164973866629569377997085, 7.65898845952016256924116370730, 8.808303348976010396664251884106, 9.420695543770157851577811017567, 10.13188262654106392802947268411

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