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Properties

Label 1-532-532.79-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.832 - 0.554i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + (0.173 + 0.984i)3-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)9-s − 11-s + (−0.766 − 0.642i)13-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.173 − 0.984i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)9-s − 11-s + (−0.766 − 0.642i)13-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.173 − 0.984i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.832 - 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1260768676 + 0.4166859813i\)
\(L(\frac12)\) \(\approx\) \(-0.1260768676 + 0.4166859813i\)
\(L(1)\) \(\approx\) \(0.6654238572 + 0.4112412349i\)
\(L(1)\) \(\approx\) \(0.6654238572 + 0.4112412349i\)

Euler product

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

−23.32282785115953551500359375314, −22.08521452001214540873994404382, −21.19664144296050093961212829016, −20.30150057190095485708706167508, −19.63570649773194993878687830917, −18.83788769945378773049880714169, −17.76927936053986312201266072995, −17.301593276887718518606737126063, −16.24141765201017783547345673263, −15.359046179201499535337440779349, −14.12658299181421229919133026557, −13.45267894867912013804753108163, −12.664418783144746822721881289969, −12.03860378489432250964885547980, −10.98246735341413574785023540958, −9.67476680164406846102849710874, −8.82838756141119149172206768752, −7.97068106737629675175692692599, −7.172552556025353070432820601554, −6.00178890345644093864721390553, −5.16372752654622007630104799865, −4.00917144130379968870477104457, −2.43989262029822522445216593295, −1.74914335081726586461429098724, −0.20183834221860377691609132855, 2.38792905479609253174483045594, 2.913742011882417593884983892440, 4.1426512359179891529781661842, 5.14654254564465685566666888886, 6.069530525686517662847756342843, 7.29887645306357436981510475849, 8.181321876402621005668070455178, 9.34640424020057440080769655249, 10.219560581920297964454953466477, 10.728907371720666473904905308901, 11.63107689141864105959173867084, 12.946479816480407386013636936650, 13.90564617637540301648948094019, 14.7757847455870857857034646707, 15.3629071349847394764416714989, 16.156378121625324955774809576248, 17.2182838798299453890505317171, 18.07279886580749654710359190585, 18.84170935945028744473284392917, 20.085117117641840881373688309208, 20.476236994452948107771675488152, 21.79383204815343142646702166754, 22.04900154271399343153451756265, 22.87574232402262045738343101096, 23.84456483016906018295547197106

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