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Properties

Label 2-864-27.7-c1-0-28
Degree $2$
Conductor $864$
Sign $0.335 + 0.941i$
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  + (1.57 − 0.713i)3-s + (0.394 + 2.23i)5-s + (−3.88 − 3.25i)7-s + (1.98 − 2.25i)9-s + (0.823 − 4.67i)11-s + (4.81 + 1.75i)13-s + (2.21 + 3.25i)15-s + (−3.30 − 5.72i)17-s + (−1.30 + 2.26i)19-s + (−8.45 − 2.37i)21-s + (−0.771 + 0.647i)23-s + (−0.154 + 0.0561i)25-s + (1.52 − 4.96i)27-s + (4.01 − 1.46i)29-s + (2.07 − 1.73i)31-s + ⋯
L(s)  = 1  + (0.911 − 0.411i)3-s + (0.176 + 1.00i)5-s + (−1.46 − 1.23i)7-s + (0.660 − 0.750i)9-s + (0.248 − 1.40i)11-s + (1.33 + 0.486i)13-s + (0.573 + 0.839i)15-s + (−0.801 − 1.38i)17-s + (−0.299 + 0.518i)19-s + (−1.84 − 0.517i)21-s + (−0.160 + 0.135i)23-s + (−0.0308 + 0.0112i)25-s + (0.292 − 0.956i)27-s + (0.745 − 0.271i)29-s + (0.371 − 0.312i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.335 + 0.941i$
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: \((2,\ 864,\ (\ :1/2),\ 0.335 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53270 - 1.08059i\)
\(L(\frac12)\) \(\approx\) \(1.53270 - 1.08059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.713i)T \)
good5 \( 1 + (-0.394 - 2.23i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.88 + 3.25i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.823 + 4.67i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-4.81 - 1.75i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.30 + 5.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 - 2.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.771 - 0.647i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.01 + 1.46i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.07 + 1.73i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.217 - 0.376i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.16 - 1.51i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.751 + 4.25i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.238 + 0.200i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-0.632 - 3.58i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.75 + 3.15i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-13.0 - 4.73i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.24 - 3.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.59 + 7.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.13 - 0.776i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (8.21 - 2.98i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (4.16 - 7.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.71 - 9.74i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.910450934585033949492510328624, −9.202994283006748093079102326629, −8.334187182508986922869224119602, −7.28732444807013135019381716400, −6.46934289625917007019426166398, −6.29302595149113876188451817076, −4.08918384477230654178850481102, −3.41082024120191076982052303347, −2.69976747288528785549558891603, −0.847313654015675730014952868279, 1.74507586356072259434503954612, 2.85972871758211775558995340911, 3.95297958046480167742086748146, 4.85932243279223305480112045818, 6.03072983462974562543059803229, 6.76914688452215518712345423653, 8.257242062477069034192728499007, 8.733300390023275384030932943813, 9.360127676869527643100659376315, 9.990987780328018017917580837025

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