Properties

Label 2-2028-13.12-c1-0-18
Degree $2$
Conductor $2028$
Sign $-0.832 + 0.554i$
Analytic cond. $16.1936$
Root an. cond. $4.02413$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4i·5-s − 2i·7-s + 9-s − 4i·11-s − 4i·15-s − 2·17-s + 2i·19-s + 2i·21-s − 11·25-s − 27-s − 6·29-s + 10i·31-s + 4i·33-s + 8·35-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78i·5-s − 0.755i·7-s + 0.333·9-s − 1.20i·11-s − 1.03i·15-s − 0.485·17-s + 0.458i·19-s + 0.436i·21-s − 2.20·25-s − 0.192·27-s − 1.11·29-s + 1.79i·31-s + 0.696i·33-s + 1.35·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 16iT - 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.813732844963216489184794391429, −7.80433978468021520564411993845, −7.12695148958217466605537996887, −6.47007881210334126460844651450, −5.92532997716241188678680318446, −4.79193386425839096467955534518, −3.54236846157479395782221387964, −3.17816743860810052265591214063, −1.72379456102557725178058192806, 0, 1.42242276437796690933912837265, 2.35866999897939738587899993063, 4.17737100000295686817537665077, 4.58728859259556129333279487065, 5.48453599620452708294071401486, 5.98465857057036710103781826432, 7.21570051869570027856824075318, 7.935730992416060941358040400705, 8.868846233846508506388452521795

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