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Properties

Label 2-864-96.11-c1-0-4
Degree $2$
Conductor $864$
Sign $-0.847 + 0.531i$
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  + (−0.649 + 1.25i)2-s + (−1.15 − 1.63i)4-s + (3.53 + 1.46i)5-s + (−2.51 + 2.51i)7-s + (2.80 − 0.393i)8-s + (−4.13 + 3.48i)10-s + (−5.60 − 2.32i)11-s + (−0.868 − 2.09i)13-s + (−1.52 − 4.78i)14-s + (−1.32 + 3.77i)16-s − 5.57·17-s + (−5.18 + 2.14i)19-s + (−1.69 − 7.45i)20-s + (6.55 − 5.53i)22-s + (0.739 − 0.739i)23-s + ⋯
L(s)  = 1  + (−0.459 + 0.888i)2-s + (−0.578 − 0.815i)4-s + (1.57 + 0.654i)5-s + (−0.948 + 0.948i)7-s + (0.990 − 0.139i)8-s + (−1.30 + 1.10i)10-s + (−1.69 − 0.700i)11-s + (−0.240 − 0.581i)13-s + (−0.407 − 1.27i)14-s + (−0.330 + 0.943i)16-s − 1.35·17-s + (−1.18 + 0.492i)19-s + (−0.379 − 1.66i)20-s + (1.39 − 1.17i)22-s + (0.154 − 0.154i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.847 + 0.531i$
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: \((2,\ 864,\ (\ :1/2),\ -0.847 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126944 - 0.441421i\)
\(L(\frac12)\) \(\approx\) \(0.126944 - 0.441421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.649 - 1.25i)T \)
3 \( 1 \)
good5 \( 1 + (-3.53 - 1.46i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.51 - 2.51i)T - 7iT^{2} \)
11 \( 1 + (5.60 + 2.32i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.868 + 2.09i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.57T + 17T^{2} \)
19 \( 1 + (5.18 - 2.14i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.739 + 0.739i)T - 23iT^{2} \)
29 \( 1 + (-0.987 - 2.38i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.29iT - 31T^{2} \)
37 \( 1 + (-0.441 + 1.06i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.383 - 0.383i)T + 41iT^{2} \)
43 \( 1 + (2.39 - 5.77i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 8.10iT - 47T^{2} \)
53 \( 1 + (-0.0479 + 0.115i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.590 + 1.42i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (6.20 - 2.56i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.495 - 1.19i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-3.91 - 3.91i)T + 71iT^{2} \)
73 \( 1 + (7.21 - 7.21i)T - 73iT^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + (-0.667 - 1.61i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-11.5 + 11.5i)T - 89iT^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−10.44349892476298744950496467680, −9.798358636668650560961934874868, −8.887520979858107170812207709836, −8.335379928111522293557439550921, −6.99051064525444498846522639113, −6.28516877287341213436753310880, −5.72166935742566803011913413212, −4.99063530097895584115275817349, −2.96040283623112401333161656067, −2.14636303904444859007238310571, 0.23022313530022111622234781383, 1.99382055837627383699014673911, 2.60939188077832848487187460018, 4.26451716160360064724746598799, 4.93643954066842187749062297090, 6.24596662012753456928950858435, 7.14231083446102767027795340077, 8.226767302559351061979754302564, 9.299253258224414679233921308075, 9.619492832648502505439491866949

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