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Properties

Label 2-864-27.7-c1-0-25
Degree $2$
Conductor $864$
Sign $0.705 + 0.708i$
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  + (0.0827 + 1.73i)3-s + (−0.532 − 3.02i)5-s + (2.03 + 1.71i)7-s + (−2.98 + 0.286i)9-s + (0.900 − 5.10i)11-s + (−0.525 − 0.191i)13-s + (5.18 − 1.17i)15-s + (−2.47 − 4.27i)17-s + (1.14 − 1.98i)19-s + (−2.79 + 3.67i)21-s + (−1.60 + 1.34i)23-s + (−4.14 + 1.50i)25-s + (−0.742 − 5.14i)27-s + (7.83 − 2.85i)29-s + (1.35 − 1.13i)31-s + ⋯
L(s)  = 1  + (0.0477 + 0.998i)3-s + (−0.238 − 1.35i)5-s + (0.770 + 0.646i)7-s + (−0.995 + 0.0954i)9-s + (0.271 − 1.54i)11-s + (−0.145 − 0.0530i)13-s + (1.33 − 0.302i)15-s + (−0.599 − 1.03i)17-s + (0.262 − 0.454i)19-s + (−0.609 + 0.800i)21-s + (−0.334 + 0.281i)23-s + (−0.829 + 0.301i)25-s + (−0.142 − 0.989i)27-s + (1.45 − 0.529i)29-s + (0.242 − 0.203i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32191 - 0.549099i\)
\(L(\frac12)\) \(\approx\) \(1.32191 - 0.549099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0827 - 1.73i)T \)
good5 \( 1 + (0.532 + 3.02i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.03 - 1.71i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.900 + 5.10i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.525 + 0.191i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.47 + 4.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.60 - 1.34i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.83 + 2.85i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.35 + 1.13i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-3.54 - 6.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.72 + 3.17i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.25 + 7.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.318 + 0.266i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 4.35T + 53T^{2} \)
59 \( 1 + (-1.47 - 8.37i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.90 + 6.63i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-7.91 - 2.88i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.71 - 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-14.8 + 5.39i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-14.3 + 5.23i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.58 + 7.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.972 - 5.51i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.919631275785432679073830236729, −8.955638768522445536768851764845, −8.644094054136824824374302176567, −7.981014870309174357931941964336, −6.34143267996095010711411327501, −5.21772876200421769269242783713, −4.92812288449790899957248264575, −3.83267158124249273492407122624, −2.59301529908939429305734427751, −0.72471477903367359221435572604, 1.56472367963419821613265523303, 2.54090366288177772486522393859, 3.82360308866385689152294503471, 4.89961475173018926518822325037, 6.46815833698227437694776507362, 6.74966432150406563437307217976, 7.69891065429706990040888343048, 8.152165068130079871454044776675, 9.508717307372396926370915662861, 10.55992532315629289348130100229

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