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Properties

Label 2-2034-339.338-c2-0-58
Degree $2$
Conductor $2034$
Sign $0.430 + 0.902i$
Analytic cond. $55.4224$
Root an. cond. $7.44462$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s − 3.08·5-s + 9.93·7-s − 2.82i·8-s − 4.36i·10-s + 6.64i·11-s − 16.2·13-s + 14.0i·14-s + 4.00·16-s − 11.4·17-s − 17.1i·19-s + 6.17·20-s − 9.40·22-s + 25.9·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s − 0.617·5-s + 1.41·7-s − 0.353i·8-s − 0.436i·10-s + 0.604i·11-s − 1.25·13-s + 1.00i·14-s + 0.250·16-s − 0.672·17-s − 0.900i·19-s + 0.308·20-s − 0.427·22-s + 1.13·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2034\)    =    \(2 \cdot 3^{2} \cdot 113\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(55.4224\)
Root analytic conductor: \(7.44462\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2034} (2033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2034,\ (\ :1),\ 0.430 + 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8549106259\)
\(L(\frac12)\) \(\approx\) \(0.8549106259\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
113 \( 1 + (98.6 + 55.1i)T \)
good5 \( 1 + 3.08T + 25T^{2} \)
7 \( 1 - 9.93T + 49T^{2} \)
11 \( 1 - 6.64iT - 121T^{2} \)
13 \( 1 + 16.2T + 169T^{2} \)
17 \( 1 + 11.4T + 289T^{2} \)
19 \( 1 + 17.1iT - 361T^{2} \)
23 \( 1 - 25.9T + 529T^{2} \)
29 \( 1 - 39.6T + 841T^{2} \)
31 \( 1 + 23.5T + 961T^{2} \)
37 \( 1 - 11.4iT - 1.36e3T^{2} \)
41 \( 1 - 13.2iT - 1.68e3T^{2} \)
43 \( 1 + 59.9iT - 1.84e3T^{2} \)
47 \( 1 + 18.9T + 2.20e3T^{2} \)
53 \( 1 - 18.3iT - 2.80e3T^{2} \)
59 \( 1 + 4.79T + 3.48e3T^{2} \)
61 \( 1 + 72.0T + 3.72e3T^{2} \)
67 \( 1 - 38.1iT - 4.48e3T^{2} \)
71 \( 1 + 78.0T + 5.04e3T^{2} \)
73 \( 1 + 93.5iT - 5.32e3T^{2} \)
79 \( 1 + 115. iT - 6.24e3T^{2} \)
83 \( 1 - 10.5iT - 6.88e3T^{2} \)
89 \( 1 - 11.0T + 7.92e3T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−8.690248468618215431341485870509, −7.88296333053008249426447473424, −7.31871871517186930541596468395, −6.74879911383817473444934277725, −5.44030252494751946638552403483, −4.67906975886014949791542760752, −4.39996887597589017683954323215, −2.88857660189745958224188323333, −1.72228190147600777748094445940, −0.23357862564690751118242112621, 1.12084623151519644108841521231, 2.17103900721223181962972549435, 3.17084059003008062672531321138, 4.27979500845896169017498876849, 4.81634980383329660641107992528, 5.65725702994331290036795217232, 6.92058971301376025132953791313, 7.82957930597257417192768251251, 8.252027147799512227244170280737, 9.085712790988977526467157737437

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