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Properties

Label 1-1323-1323.1091-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.907 + 0.419i$
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  + (0.583 + 0.811i)2-s + (−0.318 + 0.947i)4-s + (−0.797 + 0.603i)5-s + (−0.955 + 0.294i)8-s + (−0.955 − 0.294i)10-s + (0.411 − 0.911i)11-s + (−0.270 − 0.962i)13-s + (−0.797 − 0.603i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.318 − 0.947i)20-s + (0.980 − 0.198i)22-s + (0.318 − 0.947i)23-s + (0.270 − 0.962i)25-s + (0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (0.583 + 0.811i)2-s + (−0.318 + 0.947i)4-s + (−0.797 + 0.603i)5-s + (−0.955 + 0.294i)8-s + (−0.955 − 0.294i)10-s + (0.411 − 0.911i)11-s + (−0.270 − 0.962i)13-s + (−0.797 − 0.603i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.318 − 0.947i)20-s + (0.980 − 0.198i)22-s + (0.318 − 0.947i)23-s + (0.270 − 0.962i)25-s + (0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.907 + 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357849318 + 0.2989143795i\)
\(L(\frac12)\) \(\approx\) \(1.357849318 + 0.2989143795i\)
\(L(1)\) \(\approx\) \(1.030630051 + 0.4548299019i\)
\(L(1)\) \(\approx\) \(1.030630051 + 0.4548299019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.583 + 0.811i)T \)
5 \( 1 + (-0.797 + 0.603i)T \)
11 \( 1 + (0.411 - 0.911i)T \)
13 \( 1 + (-0.270 - 0.962i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.318 - 0.947i)T \)
29 \( 1 + (0.661 - 0.749i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.921 + 0.388i)T \)
43 \( 1 + (-0.797 - 0.603i)T \)
47 \( 1 + (0.995 - 0.0995i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.456 - 0.889i)T \)
61 \( 1 + (0.318 + 0.947i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.270 - 0.962i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
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

−20.81137463565301020622826570706, −20.07439581275078272877358331393, −19.57129540831169914200617597183, −19.02055514360910383698138167577, −17.836716785132951123535459471233, −17.298660388342851106052968036758, −15.95604411077969255983779816387, −15.573015589535725735997900650802, −14.653068429073126640146573422326, −13.85921434981254864617048302804, −13.0513446552760534142432788895, −12.302202085858563701451997085377, −11.64995067431760193613379855010, −11.14894423718290260030979214550, −9.989182387492080207898684259936, −9.196488969041444207830528392831, −8.65611002974318118505826626762, −7.21842735156143986168667235096, −6.68810717853840115138956014808, −5.27480759844076217085034310908, −4.62795082749244896435461042287, −4.05850186114516308092758388039, −3.01068788991686463396188442020, −1.951495127187593678135914452737, −1.0137939204623949839588360810, 0.53504566138507661643676359259, 2.53827456609205034320565413542, 3.29100347511924082006663853401, 4.07351732340181843949325132463, 4.9332953969206526607154888238, 6.05570780877138707854196536095, 6.56357814462252919736283469057, 7.56874127130720297507044088223, 8.19656564347595861081388092164, 8.870381950323253246332562942594, 10.18296065579107948748350088352, 11.0397204498791488797798809562, 11.905699298693646645407413896109, 12.48984454658356205470093620107, 13.56324135563734581051195105744, 14.129592576339647581121421573795, 15.01236089914891478622021738254, 15.52268592268500597904714250666, 16.23342283732058584373938098873, 17.05916245609561108315775205446, 17.834098063548823816371018837491, 18.658232827716550174427865570894, 19.39125177929929702184084589460, 20.32898234583500924503806220483, 21.10415801037826189165732127864

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