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Properties

Label 1-2268-2268.1103-r0-0-0
Degree $1$
Conductor $2268$
Sign $0.512 - 0.858i$
Analytic cond. $10.5325$
Root an. cond. $10.5325$
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  + (−0.597 + 0.802i)5-s + (−0.993 + 0.116i)11-s + (0.973 − 0.230i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (0.286 + 0.957i)29-s + (−0.893 − 0.448i)31-s + (0.766 − 0.642i)37-s + (0.686 + 0.727i)41-s + (−0.597 − 0.802i)43-s + (0.893 − 0.448i)47-s + (0.5 − 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (−0.597 + 0.802i)5-s + (−0.993 + 0.116i)11-s + (0.973 − 0.230i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (0.286 + 0.957i)29-s + (−0.893 − 0.448i)31-s + (0.766 − 0.642i)37-s + (0.686 + 0.727i)41-s + (−0.597 − 0.802i)43-s + (0.893 − 0.448i)47-s + (0.5 − 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(10.5325\)
Root analytic conductor: \(10.5325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (0:\ ),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6529973730 - 0.3708537983i\)
\(L(\frac12)\) \(\approx\) \(0.6529973730 - 0.3708537983i\)
\(L(1)\) \(\approx\) \(0.7979472356 + 0.05479128243i\)
\(L(1)\) \(\approx\) \(0.7979472356 + 0.05479128243i\)

Euler product

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

−19.9350053902790645427696938943, −19.00212911458340800207034740202, −18.41277395298289545664245100995, −17.7195245045082609979700270101, −16.700314987224622808560807983060, −16.16676434645307736610322628764, −15.64311366391718936055817325694, −14.88894106074511125042281089499, −13.80735694476546490263557208886, −13.2624219017642531850827828732, −12.543352023995834857863465256345, −11.81964664488972594128527584108, −11.026813959401524211834390828176, −10.37941910483023201295973149248, −9.34486106660167510668899190972, −8.59488436034163132667259871158, −8.06955194130309009161040476089, −7.28364563716649322578377848742, −6.18918752215477786009000346063, −5.552240280034196829227688947090, −4.444268226806333888811474072294, −4.10213461592039824309049805459, −2.918233456412836751690730958273, −1.981586556523384633462669548983, −0.86228068028488350561692296928, 0.30514653937520939908623388758, 1.86328662434736571137874820560, 2.62991872062428032103702042799, 3.607645465283134743994721636446, 4.19165971643067160621613760779, 5.28733537236247634582311536064, 6.16805916079967753107491722734, 6.8526071250200921584576876879, 7.73697978023545129262657759212, 8.311259738095928359039044380475, 9.15860730146228018419953560115, 10.2655182400602734025199102653, 10.86477082153043791581253354716, 11.248236214011806143191925231473, 12.32749381635524821653432121424, 13.07686931243754425144878612007, 13.67568709010645660035007507432, 14.678364566269079367692929500866, 15.27765355653524859443741466401, 15.81331876088584083504914330590, 16.52389661983936460732936862015, 17.67308922114272232355021426173, 18.165041518990687236700615845005, 18.68934669892727879947243533959, 19.707063288422237833830676469794

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