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Properties

Label 1-1480-1480.1187-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.999 + 0.000340i$
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 + (0.342 + 0.939i)13-s + (−0.342 + 0.939i)17-s + (0.173 − 0.984i)19-s + (−0.766 + 0.642i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.342 + 0.939i)33-s + (−0.173 − 0.984i)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 + (0.342 + 0.939i)13-s + (−0.342 + 0.939i)17-s + (0.173 − 0.984i)19-s + (−0.766 + 0.642i)21-s + (0.866 + 0.5i)23-s + (−0.866 − 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.342 + 0.939i)33-s + (−0.173 − 0.984i)39-s + (−0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.999 + 0.000340i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.999 + 0.000340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182722695 + 0.0002011043476i\)
\(L(\frac12)\) \(\approx\) \(1.182722695 + 0.0002011043476i\)
\(L(1)\) \(\approx\) \(0.8840049476 - 0.06113331014i\)
\(L(1)\) \(\approx\) \(0.8840049476 - 0.06113331014i\)

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 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (-0.342 + 0.939i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−20.89652791519070411215686565214, −20.13216193885315399488761749884, −18.71571799062782358600345022718, −18.42333865892512896023852708629, −17.62317499670561369694924820575, −17.10885760893651397729516962267, −16.03604734864922641748020969007, −15.417522099431616013481791518597, −14.95408700025705659527349984322, −13.73762726153515221878302631984, −12.885141131393946081006657435755, −12.04405438292760924367372734467, −11.70418127620655261872745626278, −10.56136828821057900055083303737, −10.17704356063714114030565119902, −9.12765069978747073681469706869, −8.177142955030186618379612619529, −7.35534662941207728891790000221, −6.44242396036660183872968261128, −5.48509872617453960755726927729, −5.03773525380345252385537436705, −4.17717810867726920092620513713, −2.87820891829326899133958923810, −1.902534330532409222236167696, −0.70484124576317025421230804168, 0.89150817391334026054160756108, 1.61761842587474815100814693418, 2.99494640251741933081729314685, 4.20762681410794234187567136589, 4.780547313332308885785778162951, 5.70309496594770855857570994083, 6.59508238679862803981602407581, 7.204583728816781529482734026128, 8.1827035963217714002948912669, 8.99933845145852832170079349822, 10.18944171709496635132196004901, 10.899717709193028021737655302711, 11.297988027743597747785783712, 12.110070459524518643981648402115, 13.31991217415406463939863329051, 13.487288044948858061944519359459, 14.59374356467564491195332193031, 15.590585917662249264440564904017, 16.25772771695732371424652394899, 17.04677268134415202138661736338, 17.51155725350095226508612101470, 18.330390964241645229157270845547, 19.078935317170616759220558282677, 19.77797313380560808529749029661, 20.90918827615569943451259578387

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