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Properties

Label 1-1815-1815.278-r0-0-0
Degree $1$
Conductor $1815$
Sign $-0.885 - 0.463i$
Analytic cond. $8.42882$
Root an. cond. $8.42882$
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  + (0.676 − 0.736i)2-s + (−0.0855 − 0.996i)4-s + (0.884 + 0.466i)7-s + (−0.791 − 0.610i)8-s + (−0.389 − 0.921i)13-s + (0.941 − 0.336i)14-s + (−0.985 + 0.170i)16-s + (−0.967 − 0.254i)17-s + (0.362 − 0.931i)19-s + (0.989 + 0.142i)23-s + (−0.941 − 0.336i)26-s + (0.389 − 0.921i)28-s + (−0.998 − 0.0570i)29-s + (0.516 + 0.856i)31-s + (−0.540 + 0.841i)32-s + ⋯
L(s)  = 1  + (0.676 − 0.736i)2-s + (−0.0855 − 0.996i)4-s + (0.884 + 0.466i)7-s + (−0.791 − 0.610i)8-s + (−0.389 − 0.921i)13-s + (0.941 − 0.336i)14-s + (−0.985 + 0.170i)16-s + (−0.967 − 0.254i)17-s + (0.362 − 0.931i)19-s + (0.989 + 0.142i)23-s + (−0.941 − 0.336i)26-s + (0.389 − 0.921i)28-s + (−0.998 − 0.0570i)29-s + (0.516 + 0.856i)31-s + (−0.540 + 0.841i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.885 - 0.463i$
Analytic conductor: \(8.42882\)
Root analytic conductor: \(8.42882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1815,\ (0:\ ),\ -0.885 - 0.463i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4699201409 - 1.910444392i\)
\(L(\frac12)\) \(\approx\) \(0.4699201409 - 1.910444392i\)
\(L(1)\) \(\approx\) \(1.160582492 - 0.8541448358i\)
\(L(1)\) \(\approx\) \(1.160582492 - 0.8541448358i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.676 - 0.736i)T \)
7 \( 1 + (0.884 + 0.466i)T \)
13 \( 1 + (-0.389 - 0.921i)T \)
17 \( 1 + (-0.967 - 0.254i)T \)
19 \( 1 + (0.362 - 0.931i)T \)
23 \( 1 + (0.989 + 0.142i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (-0.226 - 0.974i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.491 - 0.870i)T \)
53 \( 1 + (-0.170 + 0.985i)T \)
59 \( 1 + (0.198 - 0.980i)T \)
61 \( 1 + (-0.736 + 0.676i)T \)
67 \( 1 + (-0.909 - 0.415i)T \)
71 \( 1 + (-0.774 - 0.633i)T \)
73 \( 1 + (0.717 + 0.696i)T \)
79 \( 1 + (0.564 - 0.825i)T \)
83 \( 1 + (-0.441 - 0.897i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.999 + 0.0285i)T \)
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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

−20.79380021503439385166074220075, −19.94501731527228372052619568783, −18.867339663630549347568048442932, −18.12465669459287928733217285161, −17.27212240841955415950426458719, −16.8143376227308341403273383357, −16.122122107718742536875750046616, −14.92655557008905555137876410737, −14.84107348622850217025290698095, −13.81074695521829906486854129003, −13.33406004221785824177057754627, −12.41270065457007510714016644932, −11.51833622344555639167997001350, −11.10672489984486015650978196641, −9.8470607255709148492269690162, −8.94998750221650985722581180969, −8.14249400078731888639749647605, −7.471430503833552855390902890485, −6.70952732805192806887717862432, −5.93258325426395575191707052881, −4.85157154044740367216202884996, −4.449701996212869608312534671927, −3.546499489076687541928471231819, −2.43736850463913584143980117390, −1.4338272398384578289966016672, 0.52764043303328507786568343334, 1.72735123979184601329017615523, 2.50517214358764409798143005869, 3.29182052428148840464850615031, 4.390716304413808823306334964729, 5.13485517666011287540715093538, 5.59155865078790803190080589287, 6.79363053286233200580960932324, 7.56791143657534412479776397720, 8.8612801444783539140276969452, 9.1769883704832663516596739195, 10.483211179683214722091254804960, 10.879582339648260249585923332101, 11.71958161715717060885342497509, 12.32543304211588961979193331800, 13.20730117355115880444935071169, 13.77191815982987733298219471131, 14.687507751223891270357773162138, 15.30975076779007746964094408352, 15.75027054692118153138982060886, 17.19521724827437171050347982682, 17.78389387886538442864999227850, 18.47091235110806183657192236132, 19.28832855154176631708596358947, 20.066382205894580355906527179323

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