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Properties

Label 1-1480-1480.1277-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.981 + 0.188i$
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  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.342 + 0.939i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.342 + 0.939i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.981 + 0.188i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.981 + 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9823990245 + 0.09366644729i\)
\(L(\frac12)\) \(\approx\) \(0.9823990245 + 0.09366644729i\)
\(L(1)\) \(\approx\) \(0.7863702154 + 0.01101942444i\)
\(L(1)\) \(\approx\) \(0.7863702154 + 0.01101942444i\)

Euler product

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

−20.75757892589791625432375976926, −20.00644172530126215796261116672, −18.99260044491781960416683236299, −18.181452022770588796921237506195, −17.77529183311194228462357614078, −16.62433910812502032607399047219, −16.328138328807148128405100272463, −15.6450250870199063472558053075, −14.62815061062499358126060167729, −13.66498745607252308491466312174, −12.95271105913471645608254140840, −12.23131560571496602342931065308, −11.43939343545781797422157315391, −10.61559079761791078818118864979, −9.95132976719241484260997823522, −9.39919579370499451363512390576, −8.01414046561076086380715319391, −7.18976810363263134143447021100, −6.54812405583879781417773045466, −5.66627035619888244578774153860, −4.82396192656818253652076192568, −3.98566601197924036286112722939, −3.14278011437279560202167203977, −1.67588437410027272825985722266, −0.671100126327561874909801231843, 0.75090914451359834788633781377, 1.818301323213262842912481449068, 3.153716157787023191196544465522, 3.798211750089820737471151986398, 5.29008715224884113107957471572, 5.71153685149489727988987759234, 6.258480968884894071384630519606, 7.43440333281949455442865628411, 8.12874816417916603036851323592, 9.214477034920446931492545486024, 10.00247385860519839468282099361, 10.79510883707754823995734411064, 11.60688617553526659510569753649, 12.14858230981901503400367942808, 13.172274758882113652929650809237, 13.4695017809250055441775665647, 14.77864776669996208699185910097, 15.716849522338606647646646073059, 16.17844769011574504564178906019, 16.7728086192227966894177168097, 17.835916433808555182346623045892, 18.52598934377912794023699392187, 18.820179353395240770679595276914, 19.826527820834917278414338427546, 20.91458104320161517147102764814

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