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Properties

Label 2-2028-13.9-c1-0-25
Degree $2$
Conductor $2028$
Sign $-0.945 - 0.326i$
Analytic cond. $16.1936$
Root an. cond. $4.02413$
Motivic weight $1$
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 − 1.73·5-s + (−1.73 − 3i)7-s + (−0.499 − 0.866i)9-s + (1.73 − 3i)11-s + (−0.866 + 1.49i)15-s + (1.5 + 2.59i)17-s + (−1.73 − 3i)19-s − 3.46·21-s + (3 − 5.19i)23-s − 2.00·25-s − 0.999·27-s + (−4.5 + 7.79i)29-s + (−1.73 − 3i)33-s + (2.99 + 5.19i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.774·5-s + (−0.654 − 1.13i)7-s + (−0.166 − 0.288i)9-s + (0.522 − 0.904i)11-s + (−0.223 + 0.387i)15-s + (0.363 + 0.630i)17-s + (−0.397 − 0.688i)19-s − 0.755·21-s + (0.625 − 1.08i)23-s − 0.400·25-s − 0.192·27-s + (−0.835 + 1.44i)29-s + (−0.301 − 0.522i)33-s + (0.507 + 0.878i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6026525600\)
\(L(\frac12)\) \(\approx\) \(0.6026525600\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + (1.73 + 3i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 3i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 + 3i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (2.59 - 4.5i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.33 + 7.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (6.92 + 12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.19 - 9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.19T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.46 - 6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.602483625120644009645925385177, −7.948572085973866087575055165457, −7.01564938828947142358697466433, −6.69290456153998190410805134436, −5.63954601003475682200293348022, −4.34772762196963209432857594123, −3.66339200082767671113254953200, −2.94488765747499958760766263125, −1.33611324478509872162124031353, −0.21316177659169277573136674738, 1.87097270506334993407108986091, 2.97293056958810417057760554624, 3.78076028338625842114860770395, 4.60161467585227976025416063615, 5.60302309110354972357642328846, 6.32919021419543780314068814775, 7.47968881008694334869445476944, 7.924220137762283394105183510401, 9.053160243316872109372717367963, 9.438769004062832020100124397890

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