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Properties

Label 2-532-133.75-c1-0-7
Degree $2$
Conductor $532$
Sign $0.454 + 0.890i$
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 − 1.73i)3-s + (2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 2·13-s + (4.5 − 2.59i)17-s + (0.5 − 4.33i)19-s + (−1.00 − 5.19i)21-s + (−1.5 + 2.59i)23-s + (−2.5 − 4.33i)25-s − 4.00·27-s + (−1 − 1.73i)31-s + (3 − 5.19i)33-s + (−6 − 3.46i)37-s + (−2 − 3.46i)39-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 0.554·13-s + (1.09 − 0.630i)17-s + (0.114 − 0.993i)19-s + (−0.218 − 1.13i)21-s + (−0.312 + 0.541i)23-s + (−0.5 − 0.866i)25-s − 0.769·27-s + (−0.179 − 0.311i)31-s + (0.522 − 0.904i)33-s + (−0.986 − 0.569i)37-s + (−0.320 − 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17956 - 0.722213i\)
\(L(\frac12)\) \(\approx\) \(1.17956 - 0.722213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 4.33i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-4.5 + 2.59i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4.5 + 2.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9 + 5.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12 - 6.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3 + 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.19iT - 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.95861457792207697654114221614, −9.782483017233900771060704240788, −8.855552989618013921093741757983, −7.70401508629782014123322776292, −7.18493766863270864745978759511, −6.08335843110130534437916808931, −5.26017579815697830537039557070, −4.03685939864772390618000275178, −2.26252859629933066376413865459, −1.06702569046365590265590154517, 1.44092991793818408707059165886, 3.54021496454072951217984016544, 4.27979307852787911591790450218, 5.43558396010314325207694712242, 6.03883624365679140192858580902, 7.56574561155820761560886103482, 8.329092544735545360362192551561, 9.368272768990858295981389557635, 10.37631229043126903503113011793, 10.84272757351523736720434882453

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