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Properties

Label 1-1480-1480.499-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.995 + 0.0953i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.342 + 0.939i)13-s + (0.342 − 0.939i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.939 − 0.342i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)7-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (0.342 + 0.939i)13-s + (0.342 − 0.939i)17-s + (−0.984 − 0.173i)19-s + (0.766 − 0.642i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.939 − 0.342i)33-s + (0.984 − 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.995 + 0.0953i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.995 + 0.0953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779115677 + 0.08499003105i\)
\(L(\frac12)\) \(\approx\) \(1.779115677 + 0.08499003105i\)
\(L(1)\) \(\approx\) \(1.226744385 - 0.1331058757i\)
\(L(1)\) \(\approx\) \(1.226744385 - 0.1331058757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.642 + 0.766i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−20.799711519109437146540928926776, −19.97561967449147709531825550971, −19.365399007400686559381093589095, −18.41166398114321706227991912632, −17.27170620266373074181482502149, −16.98765082038996879845215718949, −16.22002179705356454671538962069, −15.143708484769033696308189495505, −14.79721704935924253614402680085, −13.97241267306631553681908741231, −13.21222178739196321908648955585, −12.19061729780817755371338014178, −11.01320245912347855090209199499, −10.81534540094155779188878118743, −10.061049772694244502915565750059, −8.869945549997980876192465676882, −8.42788297864568766312842943715, −7.62735695024932744842609219122, −6.348820603674129176074610286901, −5.576275074534356499603400220081, −4.70476909532767743062092287266, −3.82928529585136385959906302170, −3.28087896601936819642757672758, −1.99364603228872766448595424934, −0.73873353332924094044853051794, 1.140495004118746435143160419284, 1.96028399671769756016706145999, 2.64159638108827043268911373843, 3.938340757214209670398655304188, 4.89930802268172335282936132326, 5.804016772682330970267856827539, 6.727261652057424897466489666732, 7.35201900479523535677076845021, 8.19052734735405360593601962522, 9.07357301318747928628420646610, 9.54042570361548709217938273671, 11.206276489771854468948330923048, 11.41045939390509175818969351148, 12.36277335078182018115973556127, 12.965343137006368890700565708259, 13.89770030302581731350336543634, 14.642760368497430680130272920912, 15.05818684973208621152059319733, 16.2631029304095145211796893306, 17.16685935408584442732178125994, 17.72306419650976121247995109512, 18.52825948754790686434242628579, 19.01436081621824426429732706221, 19.809712511733256138896077434028, 20.74802258470469949682091480866

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