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Properties

Label 2-1048-131.130-c0-0-0
Degree $2$
Conductor $1048$
Sign $-i$
Analytic cond. $0.523020$
Root an. cond. $0.723201$
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  − 3-s − 7-s + 13-s + 1.41i·17-s + 1.41i·19-s + 21-s + 1.41i·23-s − 25-s + 27-s − 1.41i·31-s + 1.41i·37-s − 39-s − 41-s − 43-s + 1.41i·47-s + ⋯
L(s)  = 1  − 3-s − 7-s + 13-s + 1.41i·17-s + 1.41i·19-s + 21-s + 1.41i·23-s − 25-s + 27-s − 1.41i·31-s + 1.41i·37-s − 39-s − 41-s − 43-s + 1.41i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1048\)    =    \(2^{3} \cdot 131\)
Sign: $-i$
Analytic conductor: \(0.523020\)
Root analytic conductor: \(0.723201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1048} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1048,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4918201980\)
\(L(\frac12)\) \(\approx\) \(0.4918201980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
131 \( 1 + T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−10.14908862392410302643147367765, −9.881121358188980569282996482863, −8.592068185855285414828290204188, −7.900140989396057203826171889918, −6.62215177993869794287905606023, −5.99749659696074505737855818255, −5.56211887121719642792678103067, −4.05116597880959216005316560097, −3.33902535481590772358247225056, −1.56908270493525320923711799358, 0.52541443102369173707496324544, 2.57302160289005139429590833888, 3.64659253396289276181038913378, 4.91852525924342045121530134770, 5.60360923219166421438481674720, 6.72761292918847687038190548176, 6.86465006952430357066350568685, 8.448138664324486355840241025158, 9.081091639799794217505974475719, 10.06557127513212134046413779617

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