Properties

Label 1-532-532.131-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.371 + 0.928i$
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  + (−0.939 − 0.342i)3-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)9-s − 11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)9-s − 11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3654630232 + 0.5401128929i\)
\(L(\frac12)\) \(\approx\) \(0.3654630232 + 0.5401128929i\)
\(L(1)\) \(\approx\) \(0.7395958702 + 0.1372838287i\)
\(L(1)\) \(\approx\) \(0.7395958702 + 0.1372838287i\)

Euler product

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

−23.08339714515361113169068381276, −22.28942635909274937897350873774, −21.69184012937189193111205911539, −20.68499639531459628456203122190, −20.26113457864753994663459519055, −18.569352291326519098532017101952, −18.070410226289936805836294517888, −17.31141432174172939263576893321, −16.509251588562450678989196514379, −15.706927762989274664116785896071, −14.85052148515881908132135425821, −13.51685649786081744795211453604, −12.91566377703591824432239598402, −12.05573785871985260881526345807, −10.84110962115197163400358826024, −10.29804680828356552333784861548, −9.4562498693584905200028604603, −8.36146236871267004945341131278, −7.08342672414169206974153537092, −6.11495374945467345496006859594, −5.2396445976929382460423883103, −4.70020899021480133912334480437, −3.12278559076927050149422147135, −1.88457955581549244181236236320, −0.36855221853474078962888747563, 1.6108117677602957898018755951, 2.360246109123811482502451138448, 4.00998134810739548242799481746, 5.23486921066643395716719107945, 5.86422993791881337764963474858, 6.83061948609196665779329149934, 7.60127044010356589814290614019, 9.0324937569366053101675976608, 9.97217484363908059631206367901, 10.82804526904265198148874192734, 11.477060896573334744479430683, 12.733256056064131524401249719570, 13.27253674786834960780456994844, 14.17426880559052552243420102808, 15.3334813813340034815106369924, 16.256522388507684260462663690344, 17.16573008988914324297560644839, 17.73857377239566421081743812644, 18.57153790577641108090957214728, 19.20017680715263871954613285029, 20.57146940869462902190959954746, 21.5168369981584427969635951999, 21.96316649425713746392131335227, 22.84418576715230652023453462717, 23.90142262853388661917692537999

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