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Properties

Label 1-1480-1480.509-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.815 + 0.578i$
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  + (−0.939 − 0.342i)3-s + (−0.173 − 0.984i)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 + (−0.939 − 0.342i)19-s + (−0.173 + 0.984i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.766 + 0.642i)33-s + (0.939 − 0.342i)39-s + (0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (−0.173 − 0.984i)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 + (−0.939 − 0.342i)19-s + (−0.173 + 0.984i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.766 + 0.642i)33-s + (0.939 − 0.342i)39-s + (0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03143251122 - 0.09862558701i\)
\(L(\frac12)\) \(\approx\) \(-0.03143251122 - 0.09862558701i\)
\(L(1)\) \(\approx\) \(0.6136700846 - 0.1612211643i\)
\(L(1)\) \(\approx\) \(0.6136700846 - 0.1612211643i\)

Euler product

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

−21.34702546656518515588712566783, −20.4332496292412031226087351687, −19.59267109330089182305559363275, −18.68193049343923115511482433567, −18.052831210638587121495150084090, −17.32771090670598771228327434861, −16.677785709989570496232431959, −15.87246264261465592069973237603, −14.98613364376315338863338665495, −14.76821427147657077134449826353, −13.29764845472654323885814519378, −12.44471718664720360062855907288, −12.02566906662907946637134612722, −11.338864343728585762313167533228, −10.19079241144448047722893113802, −9.74161647692077939633519995057, −8.97188903447643328750983697035, −7.73418717118407005392996054962, −7.014849750677828258264206390194, −5.93730283728108262398332095884, −5.49078969338867794678680677791, −4.581134179199147645780931887946, −3.69149821467130886839812847168, −2.51117176032274218309479490697, −1.50500727484180762990421546911, 0.0476098390279108749302129982, 1.16826603723095065058563873581, 2.13356210080276018172188052653, 3.59236802681527908499128451491, 4.286104634586153955941263115697, 5.22867431599182998713487116092, 6.18308813196378587706619893842, 6.79525686606852884567567973578, 7.525551307952132839642757475772, 8.48182777272329216287171091832, 9.55065123561113927645078509904, 10.4612854593453826539658663318, 10.94155290260085063119863282188, 11.78070513046580621844324878506, 12.623987514294936157713825015928, 13.171639857933192768982986554217, 14.231533770358913367100992666581, 14.68005813374405550479831927785, 16.1519324829364603750615756482, 16.53524625505610312232564066368, 17.11972253168673395143320964842, 17.77277288719492612659841894555, 18.95538106929867859388326787104, 19.16820278189301628515295191233, 20.119832241648897292849523234313

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