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Properties

Label 1-1840-1840.1333-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.584 - 0.811i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + 3-s − i·7-s + 9-s − i·11-s + 13-s + i·17-s − i·19-s − i·21-s + 27-s + i·29-s + 31-s − i·33-s − 37-s + 39-s − 41-s + ⋯
L(s)  = 1  + 3-s − i·7-s + 9-s − i·11-s + 13-s + i·17-s − i·19-s − i·21-s + 27-s + i·29-s + 31-s − i·33-s − 37-s + 39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.301718693 - 1.178291906i\)
\(L(\frac12)\) \(\approx\) \(2.301718693 - 1.178291906i\)
\(L(1)\) \(\approx\) \(1.575521755 - 0.3321988196i\)
\(L(1)\) \(\approx\) \(1.575521755 - 0.3321988196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 \)
97 \( 1 \)
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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

−20.492465962546536155734765012403, −19.43892953515492808673477287925, −18.78727389825945125210960560311, −18.28185122022428571977483897787, −17.51944825912188022055674249101, −16.34689918922031477261458664513, −15.504532456668703583835285340687, −15.33118843079501019622050000732, −14.253416863939604790393936579884, −13.76967701247530226353947148260, −12.81541262237131658900303956396, −12.19986804899289389523324821987, −11.433798487838828324239113417859, −10.205376652636226381150763506070, −9.681291789108729095543211634792, −8.856504979576932409164040963753, −8.26395153381301911474858480461, −7.45706295361300965800631576421, −6.57940213078627157089227832800, −5.65705288312866603615324866209, −4.661800452461470417714186471528, −3.84418052859940448016038956924, −2.88550291783206466051725751850, −2.179370933730559312039606442309, −1.304834907504165071318960352528, 0.866020906393490619865907462402, 1.693181024993610715226904008731, 2.926344430421776432085815268698, 3.617821556018509450678027736306, 4.216103706094380646337135844531, 5.333255757538460567977842216938, 6.51386184190690835149950526983, 7.04249716717221999087733645625, 8.1497865108501429448077805807, 8.53701881517703658961871546143, 9.33247943572500384542193881413, 10.5144519575921336110073923383, 10.68714531485587274002416923823, 11.82717732286360806039605707992, 12.96570952311308453607110658315, 13.50505459367495658590454789366, 13.94123542848927105650172903027, 14.79864527044618750910742633801, 15.62348145668131522815108258313, 16.23181407069991190717570383166, 17.072663585759668474365974399190, 17.89339175204161867553291158672, 18.78271890972868916928972155178, 19.39434045724893662655230563000, 19.97862482131435658197805147311

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