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Properties

Label 2-1012-253.186-c0-0-1
Degree $2$
Conductor $1012$
Sign $-0.00711 + 0.999i$
Analytic cond. $0.505053$
Root an. cond. $0.710671$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.279 − 1.94i)3-s + (1.70 + 0.500i)5-s + (−2.74 + 0.804i)9-s + (0.841 − 0.540i)11-s + (0.496 − 3.45i)15-s + (−0.888 − 0.458i)23-s + (1.81 + 1.16i)25-s + (1.51 + 3.31i)27-s + (−0.0671 + 0.466i)31-s + (−1.28 − 1.48i)33-s + (−1.11 + 0.326i)37-s − 5.07·45-s − 1.91·47-s + (−0.142 − 0.989i)49-s + (−0.544 + 0.627i)53-s + ⋯
L(s)  = 1  + (−0.279 − 1.94i)3-s + (1.70 + 0.500i)5-s + (−2.74 + 0.804i)9-s + (0.841 − 0.540i)11-s + (0.496 − 3.45i)15-s + (−0.888 − 0.458i)23-s + (1.81 + 1.16i)25-s + (1.51 + 3.31i)27-s + (−0.0671 + 0.466i)31-s + (−1.28 − 1.48i)33-s + (−1.11 + 0.326i)37-s − 5.07·45-s − 1.91·47-s + (−0.142 − 0.989i)49-s + (−0.544 + 0.627i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00711 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00711 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.00711 + 0.999i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ -0.00711 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188492892\)
\(L(\frac12)\) \(\approx\) \(1.188492892\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
good3 \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (-1.70 - 0.500i)T + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.0671 - 0.466i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (1.11 - 0.326i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.959 - 0.281i)T^{2} \)
47 \( 1 + 1.91T + T^{2} \)
53 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-1.30 - 1.50i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.550 + 0.353i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (-0.0930 - 0.647i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \)
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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

−9.997162181805830726489878790905, −8.936169921666109624245396224043, −8.269688079101172320443251101574, −7.10162260452067024154079294723, −6.52292253539354311101784597539, −6.02029633996273565974333898486, −5.25112710616051245639626805958, −3.14419426362864153944998219114, −2.12349682963730059024146155210, −1.38283316194964296163927821638, 1.94865326503111943365003384865, 3.34591056510333896520621125559, 4.39125326629559594394449414392, 5.13476301426707322516662663881, 5.82942725847050227149352867581, 6.55933274821871183992539718574, 8.360241251030997053509197274674, 9.124031911597768486451048658810, 9.806637335008954486992573379080, 9.916327405111199686732539899886

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