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Properties

Label 2-2028-3.2-c0-0-2
Degree $2$
Conductor $2028$
Sign $1$
Analytic cond. $1.01210$
Root an. cond. $1.00603$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 2·19-s − 21-s + 25-s + 27-s − 31-s + 2·37-s − 43-s + 2·57-s − 61-s − 63-s − 67-s − 73-s + 75-s − 79-s + 81-s − 93-s − 97-s − 103-s − 109-s + 2·111-s + ⋯
L(s)  = 1  + 3-s − 7-s + 9-s + 2·19-s − 21-s + 25-s + 27-s − 31-s + 2·37-s − 43-s + 2·57-s − 61-s − 63-s − 67-s − 73-s + 75-s − 79-s + 81-s − 93-s − 97-s − 103-s − 109-s + 2·111-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.01210\)
Root analytic conductor: \(1.00603\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2028} (677, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581275760\)
\(L(\frac12)\) \(\approx\) \(1.581275760\)
\(L(1)\) 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 \( 1 - T \)
13 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−9.454574954633483056072461448238, −8.660692003001619210119893521001, −7.70476280041915015927008805123, −7.17724878266029045367539949652, −6.31527577014173187924828131402, −5.30059554267673613377994374382, −4.26356720731208039210378963644, −3.25035240254671262768438603471, −2.80967811939367674525359661562, −1.33946879406031808855362891340, 1.33946879406031808855362891340, 2.80967811939367674525359661562, 3.25035240254671262768438603471, 4.26356720731208039210378963644, 5.30059554267673613377994374382, 6.31527577014173187924828131402, 7.17724878266029045367539949652, 7.70476280041915015927008805123, 8.660692003001619210119893521001, 9.454574954633483056072461448238

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