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Properties

Label 2-532-133.10-c1-0-6
Degree $2$
Conductor $532$
Sign $0.757 + 0.652i$
Analytic cond. $4.24804$
Root an. cond. $2.06107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.317 − 1.79i)3-s + (2.76 − 0.487i)5-s + (1.96 + 1.76i)7-s + (−0.312 + 0.113i)9-s + 0.787·11-s + (3.33 + 2.79i)13-s + (−1.75 − 4.82i)15-s + (−1.10 + 3.04i)17-s + (−4.35 + 0.199i)19-s + (2.55 − 4.09i)21-s + (5.85 + 4.91i)23-s + (2.72 − 0.990i)25-s + (−2.43 − 4.21i)27-s + (−5.09 − 0.898i)29-s + (−5.06 − 8.76i)31-s + ⋯
L(s)  = 1  + (−0.183 − 1.03i)3-s + (1.23 − 0.218i)5-s + (0.743 + 0.668i)7-s + (−0.104 + 0.0379i)9-s + 0.237·11-s + (0.924 + 0.776i)13-s + (−0.453 − 1.24i)15-s + (−0.268 + 0.738i)17-s + (−0.998 + 0.0458i)19-s + (0.557 − 0.894i)21-s + (1.22 + 1.02i)23-s + (0.544 − 0.198i)25-s + (−0.468 − 0.811i)27-s + (−0.946 − 0.166i)29-s + (−0.908 − 1.57i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75020 - 0.649943i\)
\(L(\frac12)\) \(\approx\) \(1.75020 - 0.649943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.96 - 1.76i)T \)
19 \( 1 + (4.35 - 0.199i)T \)
good3 \( 1 + (0.317 + 1.79i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-2.76 + 0.487i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 0.787T + 11T^{2} \)
13 \( 1 + (-3.33 - 2.79i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.10 - 3.04i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.85 - 4.91i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (5.09 + 0.898i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (5.06 + 8.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.14 + 2.39i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.63 + 2.21i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (9.94 + 3.61i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.22 + 3.37i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.53 + 0.270i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (8.34 + 3.03i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.16 + 6.16i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.41 - 1.68i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.65 - 4.53i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.88 + 0.332i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (4.67 - 12.8i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (10.3 + 5.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.43 - 13.7i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.732 - 4.15i)T + (-91.1 + 33.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.03326734451756459753232807516, −9.643458672065517249716765908901, −8.987127836731877913481372515856, −8.085314729257364477453175956209, −6.95146244160962615000827990707, −6.10107850914392689410194012127, −5.49936464160248222723824297302, −4.05384203982370486033786721443, −2.05481628680848780263520441987, −1.58353266323813960664603986168, 1.54506033713819144577373520171, 3.15676249078125557146096023230, 4.45261931934496136301975884146, 5.16609695802949302334456938075, 6.22318585910730286915191184872, 7.21918210193565470085658648903, 8.551433221435037472924400780547, 9.300566967942060095065358443632, 10.27854031289025364069814545798, 10.71205138937984194891926500349

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