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Properties

Label 1-1480-1480.909-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.165 - 0.986i$
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  + (0.766 − 0.642i)3-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.173 − 0.984i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.173 − 0.984i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.165 - 0.986i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.165 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478033235 - 1.746713530i\)
\(L(\frac12)\) \(\approx\) \(1.478033235 - 1.746713530i\)
\(L(1)\) \(\approx\) \(1.384053156 - 0.6106477177i\)
\(L(1)\) \(\approx\) \(1.384053156 - 0.6106477177i\)

Euler product

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

−20.661536291283888720378150294573, −20.34624270073680966259152789295, −19.47813796979866159426665064652, −18.784280909162495700336835864058, −17.7866125268491808333830481889, −17.02312514062583686734609028580, −16.4140802469302109521588263456, −15.35306388217173764098711215753, −14.828566708622700426758284932859, −14.136708370235389377625703118629, −13.615478336215259503825919719685, −12.44127768615735077183352637270, −11.625625969440819931531471346277, −10.85107415095083640725211158902, −9.910704232491058774911011481959, −9.41694613102859801606141287109, −8.46535294783316709371067933479, −7.709507726092541356474369072759, −7.08406693754415302277464415233, −5.75519486419085556397332594675, −4.844629617552570665766160018692, −4.043684780324986249641390571099, −3.536942835264645606476697359532, −1.88431595453149608843041992664, −1.76380450167558610196679699371, 0.7681677841651879720771695607, 1.6670218902528996258894211331, 2.78161332543105372157888707699, 3.32248544928385014229444574666, 4.5913414616274042371674952390, 5.487721624859161686256888388763, 6.3769498907766088628603541068, 7.46295075749810930056291325196, 7.89532488686296988565996221747, 8.8592125734384826106542330308, 9.299176917099874756660636354610, 10.53161306410811157014800924119, 11.42579063287448896774018049055, 12.06311034560618997771420921246, 12.92458114757337932669909879546, 13.73589990896037032436918172490, 14.38383329573653778906424742348, 14.922569653996067814507804672709, 15.842448806534358278260664972063, 16.71123657581682983523828431959, 17.7978670352889015177899218937, 18.23049973363763130577547711258, 18.81456288577342800786229273718, 19.96162466347239762832484657035, 20.21104427316301545431848446101

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