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Properties

Label 2-2028-3.2-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.801543183\)
\(L(\frac12)\) \(\approx\) \(1.801543183\)
\(L(1)\) 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

−9.052060506000534036606390910019, −8.484211244126681244293321052855, −8.066912132663938918100343545122, −7.04921410484044153120243931167, −6.40455457469117202005815319842, −5.03809166549031321567333413416, −4.45514515562568280960611529949, −3.48892977091238303978641902382, −2.39291882477095425458968507831, −1.55492114260842589837954511465, 1.55492114260842589837954511465, 2.39291882477095425458968507831, 3.48892977091238303978641902382, 4.45514515562568280960611529949, 5.03809166549031321567333413416, 6.40455457469117202005815319842, 7.04921410484044153120243931167, 8.066912132663938918100343545122, 8.484211244126681244293321052855, 9.052060506000534036606390910019

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