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Properties

Label 2-1452-33.26-c0-0-1
Degree $2$
Conductor $1452$
Sign $0.787 - 0.615i$
Analytic cond. $0.724642$
Root an. cond. $0.851259$
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  + (−0.669 − 0.743i)3-s + (1.01 + 1.40i)5-s + (−0.104 + 0.994i)9-s + (0.360 − 1.69i)15-s + 1.73i·23-s + (−0.618 + 1.90i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)31-s + (0.309 + 0.951i)37-s + (−1.5 + 0.866i)45-s + (0.809 − 0.587i)49-s + (1.64 − 0.535i)59-s + 67-s + (1.28 − 1.15i)69-s + (−1.01 − 1.40i)71-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)3-s + (1.01 + 1.40i)5-s + (−0.104 + 0.994i)9-s + (0.360 − 1.69i)15-s + 1.73i·23-s + (−0.618 + 1.90i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)31-s + (0.309 + 0.951i)37-s + (−1.5 + 0.866i)45-s + (0.809 − 0.587i)49-s + (1.64 − 0.535i)59-s + 67-s + (1.28 − 1.15i)69-s + (−1.01 − 1.40i)71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.787 - 0.615i$
Analytic conductor: \(0.724642\)
Root analytic conductor: \(0.851259\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :0),\ 0.787 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011022614\)
\(L(\frac12)\) \(\approx\) \(1.011022614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.669 + 0.743i)T \)
11 \( 1 \)
good5 \( 1 + (-1.01 - 1.40i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.73iT - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.00160609954917241554142037719, −9.163366801169210643934251015421, −7.86984821389561805461294501870, −7.21528018452515573675329336060, −6.53667398686889951249926698190, −5.83648008721264423934051240285, −5.18360304291849945972877265506, −3.60968611628685196122883097070, −2.52050012700322518387870720049, −1.64099454667109613460972748872, 0.962500085915908788705968567549, 2.38651356999876357598051945047, 3.96106146622212769211528640427, 4.71853743851537688763101770910, 5.44680415130028421573963828017, 6.03977941500465012671721569945, 7.00843409900697899922044798111, 8.447193546928966741051088404774, 8.912019849561396151957413194948, 9.643266235442962396403488084234

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