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Properties

Label 2-532-133.121-c1-0-13
Degree $2$
Conductor $532$
Sign $-0.912 + 0.409i$
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  + (1.02 − 1.77i)3-s − 2.32·5-s + (−0.804 − 2.52i)7-s + (−0.609 − 1.05i)9-s + (−1.44 − 2.49i)11-s + (−0.126 + 0.219i)13-s + (−2.38 + 4.13i)15-s + (−2.14 + 3.71i)17-s + (−3.08 − 3.07i)19-s + (−5.30 − 1.15i)21-s + (−0.926 − 1.60i)23-s + 0.399·25-s + 3.65·27-s + (0.114 − 0.199i)29-s + (−4.46 − 7.73i)31-s + ⋯
L(s)  = 1  + (0.592 − 1.02i)3-s − 1.03·5-s + (−0.303 − 0.952i)7-s + (−0.203 − 0.351i)9-s + (−0.434 − 0.752i)11-s + (−0.0350 + 0.0607i)13-s + (−0.616 + 1.06i)15-s + (−0.519 + 0.900i)17-s + (−0.707 − 0.706i)19-s + (−1.15 − 0.252i)21-s + (−0.193 − 0.334i)23-s + 0.0798·25-s + 0.704·27-s + (0.0213 − 0.0369i)29-s + (−0.801 − 1.38i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ -0.912 + 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202160 - 0.943319i\)
\(L(\frac12)\) \(\approx\) \(0.202160 - 0.943319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.804 + 2.52i)T \)
19 \( 1 + (3.08 + 3.07i)T \)
good3 \( 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.32T + 5T^{2} \)
11 \( 1 + (1.44 + 2.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.126 - 0.219i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.14 - 3.71i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.926 + 1.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.114 + 0.199i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 + 7.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.34 + 5.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.50 - 9.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.09 - 3.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.61 + 7.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.63T + 53T^{2} \)
59 \( 1 + (0.351 - 0.608i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.19 + 8.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + (1.57 + 2.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.48 + 6.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 - 5.34T + 83T^{2} \)
89 \( 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.80 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77190909612069317551444660071, −9.427073963706286224619922104553, −8.219035205876954078239487341131, −7.908900981269931580130792961496, −7.01234797174486410887663575251, −6.16587615637386606697608460449, −4.45588020045093140429281636415, −3.57959478931039603925445838423, −2.25459845731733058026308325112, −0.50714348904975422759098615716, 2.45609162453092156430452731459, 3.55654279440074419731740476251, 4.41852493561199756847949320626, 5.39353633202998682681040387242, 6.80415577368535854649862277613, 7.83504756968586306380081868369, 8.724139142375890055864385483960, 9.380960180921460277208555307885, 10.22248624328120474478790531564, 11.12001732857196545225460228898

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