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Properties

Label 2-532-133.121-c1-0-10
Degree $2$
Conductor $532$
Sign $0.271 + 0.962i$
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  + (−1.66 + 2.88i)3-s + 2.17·5-s + (−2.16 − 1.51i)7-s + (−4.04 − 7.00i)9-s + (−3.18 − 5.51i)11-s + (1.95 − 3.39i)13-s + (−3.62 + 6.27i)15-s + (−1.57 + 2.73i)17-s + (−4.35 − 0.0546i)19-s + (7.98 − 3.72i)21-s + (−0.162 − 0.280i)23-s − 0.261·25-s + 16.9·27-s + (0.533 − 0.923i)29-s + (0.212 + 0.367i)31-s + ⋯
L(s)  = 1  + (−0.961 + 1.66i)3-s + 0.973·5-s + (−0.819 − 0.573i)7-s + (−1.34 − 2.33i)9-s + (−0.959 − 1.66i)11-s + (0.543 − 0.941i)13-s + (−0.935 + 1.62i)15-s + (−0.382 + 0.662i)17-s + (−0.999 − 0.0125i)19-s + (1.74 − 0.813i)21-s + (−0.0337 − 0.0585i)23-s − 0.0522·25-s + 3.26·27-s + (0.0990 − 0.171i)29-s + (0.0381 + 0.0660i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.271 + 0.962i$
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: \((2,\ 532,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382610 - 0.289646i\)
\(L(\frac12)\) \(\approx\) \(0.382610 - 0.289646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 1.51i)T \)
19 \( 1 + (4.35 + 0.0546i)T \)
good3 \( 1 + (1.66 - 2.88i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.17T + 5T^{2} \)
11 \( 1 + (3.18 + 5.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.95 + 3.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.57 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.162 + 0.280i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.533 + 0.923i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.212 - 0.367i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.45 - 5.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.44 + 7.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.47 + 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.71 - 4.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.446T + 53T^{2} \)
59 \( 1 + (2.60 - 4.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.453 + 0.785i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 + (0.835 + 1.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.24 + 9.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 4.07T + 83T^{2} \)
89 \( 1 + (5.04 + 8.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.29 - 3.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53579733724411675338234729967, −10.19239486039699511860432891791, −9.109440750250894762880048369194, −8.355803836096585907581038578606, −6.39162352068102916465915079933, −5.92651733557432837958445967155, −5.19312489289212766068793067730, −3.88311297555440529477642596570, −3.07558100884717998177902759950, −0.29311043589732805916046514828, 1.83931502194561046280661372531, 2.41503425808396617299864703010, 4.82750165253033356442956114676, 5.72622457044357607851820683444, 6.58434965760407078920561754716, 6.99938753321684694062495406283, 8.151213090731242480826204088722, 9.317833841309985358210602928988, 10.21088730321819118716223615411, 11.18961692913698759499405381122

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