Properties

Label 1-2736-2736.2525-r1-0-0
Degree $1$
Conductor $2736$
Sign $-0.146 - 0.989i$
Analytic cond. $294.024$
Root an. cond. $294.024$
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  + (0.342 − 0.939i)5-s + (0.5 + 0.866i)7-s − i·11-s + (−0.342 − 0.939i)13-s + (0.939 + 0.342i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + 31-s + (0.984 − 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (0.984 − 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (0.5 + 0.866i)7-s − i·11-s + (−0.342 − 0.939i)13-s + (0.939 + 0.342i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + 31-s + (0.984 − 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (0.984 − 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.146 - 0.989i$
Analytic conductor: \(294.024\)
Root analytic conductor: \(294.024\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (1:\ ),\ -0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331521676 - 1.542735001i\)
\(L(\frac12)\) \(\approx\) \(1.331521676 - 1.542735001i\)
\(L(1)\) \(\approx\) \(1.157109484 - 0.1870185688i\)
\(L(1)\) \(\approx\) \(1.157109484 - 0.1870185688i\)

Euler product

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

−19.120382716787298298010794774459, −18.69154692436635966528045740539, −17.8828537631577470794975327345, −17.08025405281791581501974185818, −16.66543101619379987397911316086, −15.76528877352897862157019721647, −14.82760969942216715857552828715, −14.229835983291047499927525308852, −13.786921315079083594323168826747, −13.13925989324471966847007287973, −11.85045837101912189708362155427, −11.36370968498487004949789991428, −10.76192674635687993751015197028, −9.912275803548219125702699083884, −9.38581268431976844135518192137, −8.249862578494270750086619461067, −7.5369316493363680953591060306, −6.93738696955744871711491406637, −6.110758204396079031661068958723, −5.34709268786538108864775611652, −4.36196382878031922235036843376, −3.50634639704799633779207436043, −2.85675106418362311831634904838, −1.74664544666112303274177029438, −0.95700367886953400060046970329, 0.3475340680177735201089260913, 1.36993298879012968581729113433, 2.15839028076618168952667036169, 2.94877590272673772808760169195, 4.2627187991770700885195721257, 4.82548513660956880077905168811, 5.64056938489259993778908304409, 6.10648341271968812443930167720, 7.52117663578975503483969490364, 7.91527658751212730060469865033, 8.85069915106552712365208308817, 9.412948571376848686982108089672, 10.17291050889392405615444695591, 10.96525122293213451870416426347, 12.06397815025168031118106811908, 12.52904915082526953948212067348, 12.85179389813868648631296531691, 14.06558139537794377692750086383, 14.651703246798817667084305322598, 15.407543147940803285727841308316, 15.97074076068952414589158516518, 17.00547412487576233049173031062, 17.40637790729458145408160306621, 18.0795126843874661944038130281, 18.85965517345688482894444024124

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