Properties

Label 1-1815-1815.1043-r0-0-0
Degree $1$
Conductor $1815$
Sign $-0.204 - 0.978i$
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  + (−0.791 + 0.610i)2-s + (0.254 − 0.967i)4-s + (0.113 − 0.993i)7-s + (0.389 + 0.921i)8-s + (0.931 − 0.362i)13-s + (0.516 + 0.856i)14-s + (−0.870 − 0.491i)16-s + (−0.717 + 0.696i)17-s + (−0.897 − 0.441i)19-s + (0.909 − 0.415i)23-s + (−0.516 + 0.856i)26-s + (−0.931 − 0.362i)28-s + (−0.985 + 0.170i)29-s + (−0.998 − 0.0570i)31-s + (0.989 − 0.142i)32-s + ⋯
L(s)  = 1  + (−0.791 + 0.610i)2-s + (0.254 − 0.967i)4-s + (0.113 − 0.993i)7-s + (0.389 + 0.921i)8-s + (0.931 − 0.362i)13-s + (0.516 + 0.856i)14-s + (−0.870 − 0.491i)16-s + (−0.717 + 0.696i)17-s + (−0.897 − 0.441i)19-s + (0.909 − 0.415i)23-s + (−0.516 + 0.856i)26-s + (−0.931 − 0.362i)28-s + (−0.985 + 0.170i)29-s + (−0.998 − 0.0570i)31-s + (0.989 − 0.142i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(8.42882\)
Root analytic conductor: \(8.42882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1815,\ (0:\ ),\ -0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4169012997 - 0.5128468631i\)
\(L(\frac12)\) \(\approx\) \(0.4169012997 - 0.5128468631i\)
\(L(1)\) \(\approx\) \(0.6816892203 + 0.02826865100i\)
\(L(1)\) \(\approx\) \(0.6816892203 + 0.02826865100i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.791 + 0.610i)T \)
7 \( 1 + (0.113 - 0.993i)T \)
13 \( 1 + (0.931 - 0.362i)T \)
17 \( 1 + (-0.717 + 0.696i)T \)
19 \( 1 + (-0.897 - 0.441i)T \)
23 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
31 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (0.633 - 0.774i)T \)
41 \( 1 + (0.564 - 0.825i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (-0.999 - 0.0285i)T \)
53 \( 1 + (0.491 + 0.870i)T \)
59 \( 1 + (-0.564 - 0.825i)T \)
61 \( 1 + (0.610 - 0.791i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (0.466 + 0.884i)T \)
73 \( 1 + (-0.676 - 0.736i)T \)
79 \( 1 + (-0.974 + 0.226i)T \)
83 \( 1 + (0.980 - 0.198i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.996 - 0.0855i)T \)
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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

−20.38551507019588335884783382416, −19.5014602271700838753253817300, −18.86856378003813956102531691223, −18.24354489337141963328062973304, −17.75944318237336729808899609949, −16.685859608683660956299544401606, −16.20721580533217136261789037503, −15.31693506991808408125890439881, −14.63701833935629111495043013193, −13.29078098128703062499543354411, −12.97145945375660184505407671206, −11.95085337726052592981800616730, −11.27209213584859824478937560978, −10.85622841664896941272780058263, −9.63101933023858685354730852828, −9.118037127319825577906424582865, −8.48861520868958912858258961764, −7.67566034967996634567709631146, −6.693105210913159352749567962179, −5.936446413910913712728543501284, −4.77822467163072137187181556792, −3.83561296650537565476472416226, −2.90411762576637873284187260934, −2.09575308936866003552146590861, −1.261838331067779560211731878972, 0.32248781321551627444186382927, 1.37002951321050928801470572910, 2.30193595657856435876504223145, 3.69802811519344085444906539903, 4.473360805451098008486475425014, 5.53396912423945986916790197235, 6.33205155674936587834854986385, 7.07451315896081305114812461839, 7.74313493823892073676672668978, 8.67494584767528735734437922152, 9.144992410988869258453486338, 10.264078282179794103655546752172, 10.969228985535743315232865756154, 11.138470467315306054444873750378, 12.8158583089074966080533048025, 13.265812681441515945362047245044, 14.29465801265048632472196354470, 14.87205147181067886525957219695, 15.67905592971258945053422204460, 16.380729067127557711602925483052, 17.170339273025011641045639590449, 17.547691642700579152798927120068, 18.45400396361212186133284946970, 19.12246082164647155066257761576, 19.92427958489250214682994157664

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