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Properties

Label 1-1480-1480.357-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.494 + 0.869i$
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  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (0.5 − 0.866i)23-s + (−0.866 − 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (0.642 − 0.766i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (0.5 − 0.866i)23-s + (−0.866 − 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (0.642 − 0.766i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.813092398 + 1.054354064i\)
\(L(\frac12)\) \(\approx\) \(1.813092398 + 1.054354064i\)
\(L(1)\) \(\approx\) \(1.305888831 + 0.4364996897i\)
\(L(1)\) \(\approx\) \(1.305888831 + 0.4364996897i\)

Euler product

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

−20.65046594770397895956381229339, −19.74596184170548534898788586934, −19.05745000389026962755407957374, −18.07747802024732469558999297435, −17.767780274070718441563825016365, −17.12482294323065706664611596359, −15.88703358029281089545720008925, −15.02578430577771124962167911523, −14.55405742180357297557276796672, −13.51969740988522112496334286108, −13.09028314366185183172869621559, −12.20772628878838032942898756605, −11.429983572017900890208012574993, −10.69598301831363826055480080793, −9.677183301630840399764620979646, −8.697188476652052687174680280034, −7.91348328786404423515559491866, −7.50717777437961511308196100621, −6.532022007170533918108964042487, −5.557228007352637421328235114670, −4.792001063273691245548729326253, −3.58164819464147211437252646267, −2.68431454579312644174254451405, −1.681110128312914548614686810453, −0.95104060095634097070188903534, 1.08172174743285821874613294529, 2.304409451136050419640179464119, 3.233202340469416703800276779581, 4.023221997659836742864606082427, 5.02977279983970578266250714883, 5.50959769590937357972924130332, 6.62584129412723664067772670332, 7.95790055980766433172697502110, 8.35804088882707459877064453660, 9.11660545636705615655889050147, 10.10162836001045022991384438560, 10.79338246615882494740068589347, 11.4441040343382565705400121395, 12.21664773005023805008967357126, 13.50352491599658977822505857257, 14.23102524479519045676243908961, 14.50994076459467980754356089577, 15.70305599285240357664272329892, 16.13447965462941564460076312867, 16.836086274276870088392894200639, 17.78297460725641271179525184879, 18.69973339803815484084203979608, 19.136974406084739238731352731547, 20.442671143431617019759266719708, 20.833119999861400755861431954274

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