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Properties

Label 2-532-133.10-c1-0-10
Degree $2$
Conductor $532$
Sign $0.614 + 0.789i$
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  + (−0.0621 − 0.352i)3-s + (2.12 − 0.375i)5-s + (0.943 − 2.47i)7-s + (2.69 − 0.982i)9-s − 3.77·11-s + (0.242 + 0.203i)13-s + (−0.264 − 0.726i)15-s + (−0.714 + 1.96i)17-s + (2.16 − 3.78i)19-s + (−0.929 − 0.179i)21-s + (−2.05 − 1.72i)23-s + (−0.315 + 0.114i)25-s + (−1.05 − 1.81i)27-s + (3.99 + 0.703i)29-s + (1.54 + 2.67i)31-s + ⋯
L(s)  = 1  + (−0.0358 − 0.203i)3-s + (0.951 − 0.167i)5-s + (0.356 − 0.934i)7-s + (0.899 − 0.327i)9-s − 1.13·11-s + (0.0672 + 0.0564i)13-s + (−0.0682 − 0.187i)15-s + (−0.173 + 0.476i)17-s + (0.497 − 0.867i)19-s + (−0.202 − 0.0390i)21-s + (−0.429 − 0.360i)23-s + (−0.0631 + 0.0229i)25-s + (−0.202 − 0.350i)27-s + (0.741 + 0.130i)29-s + (0.276 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.614 + 0.789i$
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: \((2,\ 532,\ (\ :1/2),\ 0.614 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54225 - 0.754209i\)
\(L(\frac12)\) \(\approx\) \(1.54225 - 0.754209i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.943 + 2.47i)T \)
19 \( 1 + (-2.16 + 3.78i)T \)
good3 \( 1 + (0.0621 + 0.352i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-2.12 + 0.375i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 + (-0.242 - 0.203i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.714 - 1.96i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.05 + 1.72i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.99 - 0.703i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.90 + 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.706 - 0.592i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-6.24 - 2.27i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.14 + 3.13i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-3.30 - 0.583i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.65 - 0.968i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (4.20 - 5.01i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.11 - 2.51i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (4.60 - 12.6i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (4.27 - 0.754i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.50 + 4.14i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.75 + 2.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.83 - 10.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.274 - 1.55i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.42175940706883792264450249064, −10.08289839993009933893530041402, −9.030567277553499824188771583319, −7.890421993999858765826335178171, −7.12271916953063460239419258758, −6.15105802450847708061221871018, −5.04943293040391798184501782820, −4.10117870801467737045561205187, −2.49252063317637271510752330857, −1.13412173912734363329494866366, 1.81543498355421478688908170145, 2.82349342585913581091197939517, 4.49918385040114631508057463357, 5.45158140162686789293876871673, 6.13852162723891706777829119555, 7.49423365837652732284535460832, 8.240407980696228809906253775706, 9.479818940332242559464980569337, 9.982683377798602126464191940760, 10.79574462249709777922731848443

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