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Properties

Label 2-2028-39.23-c0-0-1
Degree $2$
Conductor $2028$
Sign $0.252 - 0.967i$
Analytic cond. $1.01210$
Root an. cond. $1.00603$
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  + (0.5 + 0.866i)3-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + 1.73i·21-s − 25-s − 0.999·27-s − 1.73i·31-s + (0.5 − 0.866i)43-s + (1 + 1.73i)49-s + (−0.5 + 0.866i)61-s + (−1.49 + 0.866i)63-s + (−1.5 + 0.866i)67-s − 1.73i·73-s + (−0.5 − 0.866i)75-s + 79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + 1.73i·21-s − 25-s − 0.999·27-s − 1.73i·31-s + (0.5 − 0.866i)43-s + (1 + 1.73i)49-s + (−0.5 + 0.866i)61-s + (−1.49 + 0.866i)63-s + (−1.5 + 0.866i)67-s − 1.73i·73-s + (−0.5 − 0.866i)75-s + 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(1.01210\)
Root analytic conductor: \(1.00603\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552350892\)
\(L(\frac12)\) \(\approx\) \(1.552350892\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.284804723633816577482141140160, −8.844173110394472781463167415039, −7.948950295788497112579356918428, −7.60706026699309171868249270364, −6.03454453587760628266545405999, −5.39871823781162242755620560095, −4.59673344106492540293986117982, −3.87865735450143787661130835212, −2.60803587655338948401594424580, −1.85710015148427093512776057106, 1.19736238697406130626784494067, 1.99761782792138409732538620945, 3.24547977871478479452233466290, 4.24421059907432348318115350246, 5.10402976246783382971351336778, 6.13655952796112925254041787196, 7.07046159732048141288919066605, 7.65878522778450404790269942355, 8.230181683718558244925416899976, 8.919463747275911891923516760064

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