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Properties

Label 2-1032-1032.323-c0-0-1
Degree $2$
Conductor $1032$
Sign $0.996 - 0.0789i$
Analytic cond. $0.515035$
Root an. cond. $0.717659$
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  + (−0.900 − 0.433i)2-s + (0.900 + 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.623 − 0.781i)6-s + (−0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.678 − 0.541i)11-s + (0.222 + 0.974i)12-s + (−0.222 + 0.974i)16-s + (1.52 + 0.347i)17-s + (−0.222 − 0.974i)18-s + (1.22 − 0.974i)19-s + (0.376 + 0.781i)22-s + (0.222 − 0.974i)24-s + (−0.900 + 0.433i)25-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.900 + 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.623 − 0.781i)6-s + (−0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + (−0.678 − 0.541i)11-s + (0.222 + 0.974i)12-s + (−0.222 + 0.974i)16-s + (1.52 + 0.347i)17-s + (−0.222 − 0.974i)18-s + (1.22 − 0.974i)19-s + (0.376 + 0.781i)22-s + (0.222 − 0.974i)24-s + (−0.900 + 0.433i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $0.996 - 0.0789i$
Analytic conductor: \(0.515035\)
Root analytic conductor: \(0.717659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1032} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1032,\ (\ :0),\ 0.996 - 0.0789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9499131993\)
\(L(\frac12)\) \(\approx\) \(0.9499131993\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 + 0.433i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
good5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
19 \( 1 + (-1.22 + 0.974i)T + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.623 - 0.781i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (0.846 + 0.193i)T + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.623 - 0.781i)T^{2} \)
67 \( 1 + (1.24 + 1.56i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
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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

−9.813637749065337495439553404226, −9.562947233458316358396443132525, −8.516871371029264294730767636337, −7.84021032826312583424492531736, −7.34122848703402983370246788642, −5.94450009630517077285406120697, −4.74953979319631252398860735829, −3.36739551624339211732114896537, −2.95026666069965826172643709647, −1.50606128339238078074281784438, 1.35968865097591052526899113013, 2.51200909976650606488573433504, 3.62650569492538331332359892263, 5.21234158338583292916392010383, 6.03395573910694938474056392572, 7.28855206432871967268185309678, 7.61273189689678975661454025046, 8.317184172148591416665012457503, 9.298176439260503121578597525385, 9.933977677188660666651430230618

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