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Properties

Label 2-532-133.10-c1-0-0
Degree $2$
Conductor $532$
Sign $-0.969 + 0.244i$
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  + (0.438 + 2.48i)3-s + (−1.03 + 0.181i)5-s + (−2.55 − 0.670i)7-s + (−3.18 + 1.15i)9-s − 2.06·11-s + (−0.478 − 0.401i)13-s + (−0.905 − 2.48i)15-s + (−1.43 + 3.93i)17-s + (−3.76 + 2.19i)19-s + (0.544 − 6.66i)21-s + (−4.26 − 3.57i)23-s + (−3.66 + 1.33i)25-s + (−0.490 − 0.849i)27-s + (7.92 + 1.39i)29-s + (−0.252 − 0.437i)31-s + ⋯
L(s)  = 1  + (0.253 + 1.43i)3-s + (−0.461 + 0.0813i)5-s + (−0.967 − 0.253i)7-s + (−1.06 + 0.386i)9-s − 0.622·11-s + (−0.132 − 0.111i)13-s + (−0.233 − 0.642i)15-s + (−0.347 + 0.955i)17-s + (−0.864 + 0.503i)19-s + (0.118 − 1.45i)21-s + (−0.888 − 0.745i)23-s + (−0.733 + 0.266i)25-s + (−0.0944 − 0.163i)27-s + (1.47 + 0.259i)29-s + (−0.0453 − 0.0786i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.969 + 0.244i$
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: \((2,\ 532,\ (\ :1/2),\ -0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0700646 - 0.565070i\)
\(L(\frac12)\) \(\approx\) \(0.0700646 - 0.565070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 + 0.670i)T \)
19 \( 1 + (3.76 - 2.19i)T \)
good3 \( 1 + (-0.438 - 2.48i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (1.03 - 0.181i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 + 2.06T + 11T^{2} \)
13 \( 1 + (0.478 + 0.401i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.43 - 3.93i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.26 + 3.57i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.92 - 1.39i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.252 + 0.437i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.07 + 3.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.356 + 0.299i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (8.38 + 3.05i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.662 - 1.82i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.276 + 0.0487i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-6.12 - 2.22i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.44 - 8.87i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.76 - 4.49i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.35 - 9.22i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-9.01 + 1.58i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (4.23 - 11.6i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.36 - 2.51i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.04 - 11.6i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.08 - 6.16i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.91450135807758058240406264881, −10.29637850464245872477426500278, −9.829074792895999453101983193431, −8.721436772052140842508576744793, −8.020599178216914503912041050177, −6.67631404759489516036609637744, −5.67572784670606821691637998282, −4.34016776057669541120221196094, −3.83283540440384591990145126415, −2.66227270777722281946690513437, 0.30033857921307907045958076959, 2.14902185596829589340890210706, 3.11989778218925495553673982889, 4.65428260525579214164233216974, 6.12070638785765771309062178785, 6.72466482896591569889367499955, 7.68159231831007945965504554850, 8.304914520066700042915913001468, 9.375217016277851354285332202965, 10.31311277940106164443683036223

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