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Properties

Label 2-532-133.11-c1-0-7
Degree $2$
Conductor $532$
Sign $0.0622 - 0.998i$
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  + (1.58 + 2.75i)3-s + 2.85·5-s + (1.33 + 2.28i)7-s + (−3.54 + 6.14i)9-s + (0.978 − 1.69i)11-s + (−2.81 − 4.86i)13-s + (4.52 + 7.84i)15-s + (0.0951 + 0.164i)17-s + (−1.31 − 4.15i)19-s + (−4.15 + 7.30i)21-s + (2.03 − 3.52i)23-s + 3.12·25-s − 13.0·27-s + (−3.37 − 5.84i)29-s + (2.76 − 4.78i)31-s + ⋯
L(s)  = 1  + (0.917 + 1.58i)3-s + 1.27·5-s + (0.505 + 0.862i)7-s + (−1.18 + 2.04i)9-s + (0.295 − 0.511i)11-s + (−0.779 − 1.35i)13-s + (1.16 + 2.02i)15-s + (0.0230 + 0.0399i)17-s + (−0.300 − 0.953i)19-s + (−0.906 + 1.59i)21-s + (0.423 − 0.734i)23-s + 0.624·25-s − 2.50·27-s + (−0.626 − 1.08i)29-s + (0.496 − 0.860i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0622 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.0622 - 0.998i$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ 0.0622 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71011 + 1.60674i\)
\(L(\frac12)\) \(\approx\) \(1.71011 + 1.60674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 2.28i)T \)
19 \( 1 + (1.31 + 4.15i)T \)
good3 \( 1 + (-1.58 - 2.75i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.85T + 5T^{2} \)
11 \( 1 + (-0.978 + 1.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.81 + 4.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.0951 - 0.164i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.03 + 3.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.37 + 5.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.76 + 4.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.08 + 1.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.78 - 10.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.76 - 3.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.322 - 0.558i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.986T + 53T^{2} \)
59 \( 1 + (-0.517 - 0.896i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.63 - 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 2.98T + 67T^{2} \)
71 \( 1 + (7.31 - 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.60 + 2.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 1.00T + 83T^{2} \)
89 \( 1 + (-3.33 + 5.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.455 - 0.789i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.72314756330852697939420789044, −9.965370673301687135988805875875, −9.403935993337369105546862174653, −8.659914257849886059524259452573, −7.903804305381126492364265594703, −6.08459246494268131365022120807, −5.28652644039624607935283778427, −4.52105786983106785535757846195, −2.98478209976894718050678025018, −2.35671507711614687214481510448, 1.61591283019238957079251268211, 1.95556623915889385782767018559, 3.54088342822400835745324628663, 5.10061552156946356607687691985, 6.47144636797934281639128793836, 6.99494733219979858426248760337, 7.75568618363911135200641007907, 8.870172072364436492261012996200, 9.497394581158860719693242673855, 10.50718795807322518410049657967

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