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Properties

Label 1-1323-1323.1004-r0-0-0
Degree $1$
Conductor $1323$
Sign $-0.0901 + 0.995i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (0.411 + 0.911i)2-s + (−0.661 + 0.749i)4-s + (0.921 − 0.388i)5-s + (−0.955 − 0.294i)8-s + (0.733 + 0.680i)10-s + (0.583 − 0.811i)11-s + (−0.270 + 0.962i)13-s + (−0.124 − 0.992i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.318 + 0.947i)20-s + (0.980 + 0.198i)22-s + (−0.980 − 0.198i)23-s + (0.698 − 0.715i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.661 + 0.749i)4-s + (0.921 − 0.388i)5-s + (−0.955 − 0.294i)8-s + (0.733 + 0.680i)10-s + (0.583 − 0.811i)11-s + (−0.270 + 0.962i)13-s + (−0.124 − 0.992i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.318 + 0.947i)20-s + (0.980 + 0.198i)22-s + (−0.980 − 0.198i)23-s + (0.698 − 0.715i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0901 + 0.995i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ -0.0901 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441748056 + 1.578087574i\)
\(L(\frac12)\) \(\approx\) \(1.441748056 + 1.578087574i\)
\(L(1)\) \(\approx\) \(1.253314153 + 0.7303283862i\)
\(L(1)\) \(\approx\) \(1.253314153 + 0.7303283862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.411 + 0.911i)T \)
5 \( 1 + (0.921 - 0.388i)T \)
11 \( 1 + (0.583 - 0.811i)T \)
13 \( 1 + (-0.270 + 0.962i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.980 - 0.198i)T \)
29 \( 1 + (0.661 + 0.749i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.921 - 0.388i)T \)
43 \( 1 + (-0.797 + 0.603i)T \)
47 \( 1 + (-0.583 + 0.811i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (0.542 - 0.840i)T \)
61 \( 1 + (0.661 + 0.749i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.365 + 0.930i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.270 + 0.962i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−20.66456294125428654717445990213, −20.10681303008499838735355617343, −19.45483743963392888435372871532, −18.36024163407644384823897206931, −17.841026864386714356032406654914, −17.37095016538276414312621420858, −16.053649206224680957520824629379, −15.06172455148949244828444276568, −14.46180489967253976354186007465, −13.68654397352184215291957249526, −13.11270113723030797459576932271, −12.155339815456314662586853093367, −11.56607967725754041746613159594, −10.55073916330442431026287459671, −9.79480440366229150800969226861, −9.51987420239668145117524561359, −8.32227298033480993732311774201, −7.10521308462615603439842332907, −6.27028211878997622131850200835, −5.294419117136082922801117650251, −4.74031651174405937054379445457, −3.48818075680908713814011076515, −2.70776547492179057937197962873, −1.92934374037767208711775492401, −0.837867962176303374648077824605, 1.15130962637564084870668943833, 2.31986863897067097901613245814, 3.60020569761849917082680073757, 4.30422537602742886491234886536, 5.35812291408339556672295177217, 6.08282078396908094320594610300, 6.54239356496899182682800760532, 7.77310878306689711883293091619, 8.47222252302741259432554489663, 9.32985203070611584835283802950, 9.896201980773144628764449185501, 11.16433288057801605183728597398, 12.20747411434374205199311806656, 12.742593159634594803514680573, 13.764979372407873643750451246268, 14.227539087956416103119309483376, 14.75485860211868256494979946062, 16.127091496071136474071622302230, 16.46659156653683074839506024148, 17.11947831308528731343496188674, 17.91663857433544108504020007659, 18.65062622756648153913946866393, 19.551001993428552196882355438580, 20.65949109169296450569883581141, 21.44491338262415674524585886635

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