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Properties

Label 1-456-456.275-r0-0-0
Degree $1$
Conductor $456$
Sign $0.934 + 0.356i$
Analytic cond. $2.11765$
Root an. cond. $2.11765$
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  + (0.766 + 0.642i)5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.173 + 0.984i)35-s − 37-s + (−0.173 + 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)5-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.173 + 0.984i)25-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.173 + 0.984i)35-s − 37-s + (−0.173 + 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(2.11765\)
Root analytic conductor: \(2.11765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (0:\ ),\ 0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676260564 + 0.3092483523i\)
\(L(\frac12)\) \(\approx\) \(1.676260564 + 0.3092483523i\)
\(L(1)\) \(\approx\) \(1.313596435 + 0.1471406430i\)
\(L(1)\) \(\approx\) \(1.313596435 + 0.1471406430i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−23.99898508021648911484882595247, −23.13726200693840958302978735228, −22.16746752328774676906123203448, −21.0427577249337683803685449836, −20.755126621354576130436827375621, −19.71848289117456475860511666950, −18.79562002254516810246674296352, −17.5496109041099080178381842411, −17.10503810762459522613468402394, −16.436110820332585105677807353911, −15.06275216166649145786800544223, −14.21803634956180716238523281775, −13.52792455401825085450629676341, −12.522072374490702149849137536322, −11.66039705315675544709064329567, −10.50093346448349671004640993420, −9.64333713746424379156915998086, −8.89558197173048484386940526363, −7.623385339831769227414591591343, −6.8285011074419559877393057303, −5.58229203891195494571074222822, −4.651723762762745823854125518157, −3.760200998812567775806548271517, −2.01812012518085359522523048765, −1.247914404994298273408971368814, 1.29712195549511007872339393833, 2.61274464088325685869044305421, 3.36040289478596908938621705004, 5.09520496524504234063584200135, 5.74716132184140916841520737111, 6.70661490926281867611897011344, 7.92885596751241299644313564846, 8.84960102926208165689257689092, 9.81407919434310303892514004583, 10.75975883794023478154698246185, 11.60096959910529201858187520729, 12.614914345726893731907566996430, 13.62905389184613032765841753140, 14.55361206756234493566579676608, 15.05907349429523074293432157859, 16.2832223796082702811957118103, 17.23090594428643356080987221557, 18.07009175937241175895716824336, 18.72502727935152435402835318355, 19.56214223286333700366388939405, 20.984377665211563827220330652040, 21.2943663310246831497463107055, 22.39359890167699761731264791929, 22.81994387392017149950580232905, 24.27698049400358124321738444187

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