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Properties

Label 1-465-465.242-r0-0-0
Degree $1$
Conductor $465$
Sign $0.816 - 0.577i$
Analytic cond. $2.15945$
Root an. cond. $2.15945$
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  − i·2-s − 4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + 29-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(2.15945\)
Root analytic conductor: \(2.15945\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 465,\ (0:\ ),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9722329455 - 0.3093423487i\)
\(L(\frac12)\) \(\approx\) \(0.9722329455 - 0.3093423487i\)
\(L(1)\) \(\approx\) \(0.8398336854 - 0.3311239131i\)
\(L(1)\) \(\approx\) \(0.8398336854 - 0.3311239131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good2 \( 1 - iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + iT \)
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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

−24.22399533619608669432471001426, −23.03128911378688203782579697569, −22.448323154152440452567988302243, −21.80980688139685060114681282587, −20.632139198876334965328739367593, −19.258404206735939379546577761409, −18.90516186671110070167784758793, −17.85624080429982117307107076046, −16.843993859623710184679450302219, −16.24182757741955069032797274429, −15.47903906598884789003609771794, −14.386366583878020266065179736661, −13.872127687267387976042669787780, −12.57173639458931353007176472186, −12.10638696551636524645961764248, −10.40398604691028648943683337359, −9.615180837171110396657772997512, −8.735030603095512390075982152524, −7.80748337744913096463569883034, −6.77692125264339115348707990903, −5.90809819144590899289325514506, −5.14373955780828410204663293922, −3.77884678809861920096522450552, −2.82742024252839607521723233870, −0.73843264581849637382546737246, 1.05603058239151500335569611416, 2.33659620531022492683132352168, 3.42429745329091272941523820799, 4.317297815449040487860254849464, 5.35660710101200215808095882542, 6.76450369124119947295696288323, 7.68563064350517383142846699081, 9.11051052272399294996578333439, 9.73779354317884756303168091817, 10.42246970264229610862383830069, 11.66857153716312043020173930303, 12.31017873550414314624613962237, 13.17444519773188692867609272183, 14.05840300891069923951529439367, 14.90485227697336200791361206865, 16.17601365753880566598246072616, 17.22474156385797152564538289164, 17.755344771376302600630278538990, 19.06646752860281196493824950670, 19.60728451221053521970059147337, 20.1702357381410495375071567152, 21.348908385073278486761067630679, 21.94968827175458523751737530239, 22.92818184467929301838805738508, 23.38910841226351465317188670976

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