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Properties

Label 1-648-648.283-r1-0-0
Degree $1$
Conductor $648$
Sign $-0.963 + 0.268i$
Analytic cond. $69.6372$
Root an. cond. $69.6372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.893 − 0.448i)5-s + (0.286 + 0.957i)7-s + (−0.0581 + 0.998i)11-s + (−0.597 + 0.802i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.286 − 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (0.686 + 0.727i)31-s + (0.173 − 0.984i)35-s + (−0.173 − 0.984i)37-s + (−0.993 − 0.116i)41-s + (−0.835 − 0.549i)43-s + (0.686 − 0.727i)47-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)5-s + (0.286 + 0.957i)7-s + (−0.0581 + 0.998i)11-s + (−0.597 + 0.802i)13-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.286 − 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (0.686 + 0.727i)31-s + (0.173 − 0.984i)35-s + (−0.173 − 0.984i)37-s + (−0.993 − 0.116i)41-s + (−0.835 − 0.549i)43-s + (0.686 − 0.727i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.963 + 0.268i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ -0.963 + 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09917152778 + 0.7260640362i\)
\(L(\frac12)\) \(\approx\) \(0.09917152778 + 0.7260640362i\)
\(L(1)\) \(\approx\) \(0.8261543087 + 0.1898860235i\)
\(L(1)\) \(\approx\) \(0.8261543087 + 0.1898860235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.893 - 0.448i)T \)
7 \( 1 + (0.286 + 0.957i)T \)
11 \( 1 + (-0.0581 + 0.998i)T \)
13 \( 1 + (-0.597 + 0.802i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.286 - 0.957i)T \)
29 \( 1 + (-0.396 + 0.918i)T \)
31 \( 1 + (0.686 + 0.727i)T \)
37 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (-0.993 - 0.116i)T \)
43 \( 1 + (-0.835 - 0.549i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.0581 - 0.998i)T \)
61 \( 1 + (-0.973 - 0.230i)T \)
67 \( 1 + (0.396 + 0.918i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.893 - 0.448i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.43543934450383156880107366238, −21.50352121715624609626059959095, −20.55146501846588505848087342823, −19.719070017415553521951857866795, −19.183513540689908319708655831189, −18.24696888029955483739371458044, −17.2059316729451451544135892988, −16.61682237125563109137639011389, −15.48420470233418560326893944757, −14.95987407916843656977539177805, −13.8623588363502300698150968128, −13.2566816656713435449573632329, −11.96649659423467805442691886750, −11.33084335327986877686142049241, −10.50734818837556540753883989484, −9.6948748090497552348782683871, −8.20215557630451020447345616800, −7.78281490318867195103850041587, −6.89316540839760930911511885530, −5.71524659375090626341213509212, −4.63946831927812887801916769596, −3.56767209062078954808632640022, −2.94828866707502416608612870726, −1.16827199620833327698634903425, −0.19646702668997590391933359354, 1.34472075325502115044754626996, 2.48531833264975620070834244253, 3.66311276476646393045969398483, 4.82411531820305223133885162896, 5.31307792091901067188009819818, 6.8449037829966609680649585586, 7.55568704285729591196993089021, 8.55976310291473584819757183163, 9.29294870760584628014998841654, 10.27526149377615276773532811035, 11.555928379217482961430336158401, 12.17506256043496764937797230757, 12.5684603515081686326443939061, 14.07989653662028714762026441251, 14.78486268448601216923223157771, 15.58597505288562256793943486422, 16.36092593737387118755630495659, 17.1480624369549494018524666607, 18.42304163522052043408384653351, 18.731515131685494427880166840262, 19.88252894983964849240245134601, 20.51069133410282742646426792694, 21.33849249872863672474309894816, 22.30478447661496845728091430523, 23.077315981938520140226931237757

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