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Properties

Label 1-285-285.158-r0-0-0
Degree $1$
Conductor $285$
Sign $0.794 + 0.607i$
Analytic cond. $1.32353$
Root an. cond. $1.32353$
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.642 + 0.766i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + (0.642 − 0.766i)22-s + (−0.342 + 0.939i)23-s + (0.5 + 0.866i)26-s + (−0.984 + 0.173i)28-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.642 + 0.766i)32-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.642 + 0.766i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + (0.642 − 0.766i)22-s + (−0.342 + 0.939i)23-s + (0.5 + 0.866i)26-s + (−0.984 + 0.173i)28-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.642 + 0.766i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ 0.794 + 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.123652675 + 0.7191127220i\)
\(L(\frac12)\) \(\approx\) \(2.123652675 + 0.7191127220i\)
\(L(1)\) \(\approx\) \(1.793257232 + 0.3624032395i\)
\(L(1)\) \(\approx\) \(1.793257232 + 0.3624032395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−25.39301217715237163826448168364, −24.523242443319562226798877864530, −23.17187412115006274812206255589, −22.98114083280864139299268814007, −22.04453781297238168893527550373, −20.88970363995177163587748518240, −20.17690069397029811640546081767, −19.44384103290165956625462973441, −18.27511411818136997882932010008, −16.91053750010799407230168236203, −16.14790651796526231084997178283, −15.167446751236997632673735793647, −14.2859325714052834348898740309, −13.271473954111070604953554545258, −12.56409802564329125930910624955, −11.64356817743709828006253817156, −10.363661617844621848197264720166, −9.78608568321452305058262862119, −8.05514551887884174306213516048, −6.88179536136881429303450795158, −6.12734503543745834768294609992, −4.87023215416604953500923447041, −3.77065358859916395586189971365, −2.88319301065470856194668725783, −1.31354196387572555455282136075, 1.7164741372738720822029407437, 3.22518598508514439579827068239, 3.84654495881828976339886186928, 5.4401405464026972028950737675, 6.13749855565498200782464469204, 7.1124737024899657602272388623, 8.433026943652344513014958663997, 9.550534641140158424915513911327, 10.91406919915429161419236828139, 11.813029926029411716857867003, 12.65840263074260032495597485412, 13.66841171187432341694682939740, 14.37501170409342677774280485610, 15.54914578086200704289250662235, 16.2805139818097246620519493786, 16.988229335614713565912200917340, 18.60536571239104562042171775475, 19.351395778086738004474073506774, 20.35451499350566117132385877507, 21.565936975435039140140073348985, 21.86727947403404183767861702899, 23.01381426182726735354236896912, 23.71917484915041380209186856276, 24.643287119058137918598568706782, 25.61223124997490477989195955204

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