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Properties

Label 2-2028-13.9-c1-0-15
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 + 4.14·5-s + (−1.20 − 2.08i)7-s + (−0.499 − 0.866i)9-s + (1.73 − 3i)11-s + (−2.07 + 3.58i)15-s + (2.58 + 4.48i)17-s + (−1.73 − 3i)19-s + 2.41·21-s + (−1 + 1.73i)23-s + 12.1·25-s + 0.999·27-s + (−1.58 + 2.75i)29-s + 1.05·31-s + (1.73 + 3i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 1.85·5-s + (−0.455 − 0.789i)7-s + (−0.166 − 0.288i)9-s + (0.522 − 0.904i)11-s + (−0.535 + 0.926i)15-s + (0.628 + 1.08i)17-s + (−0.397 − 0.688i)19-s + 0.526·21-s + (−0.208 + 0.361i)23-s + 2.43·25-s + 0.192·27-s + (−0.295 + 0.511i)29-s + 0.188·31-s + (0.301 + 0.522i)33-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

\(L(1)\) \(\approx\) \(2.229198472\)
\(L(\frac12)\) \(\approx\) \(2.229198472\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.14T + 5T^{2} \)
7 \( 1 + (1.20 + 2.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 3i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.58 - 4.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 + 3i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 - 2.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.05T + 31T^{2} \)
37 \( 1 + (-3.80 + 6.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.340 + 0.589i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.08 + 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 + (-5.87 - 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.20 + 2.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 + 3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 + 1.36T + 83T^{2} \)
89 \( 1 + (-3.46 + 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.81 - 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.056043901812262887443155033021, −8.785685681373790416726773589578, −7.34827461705914988263577064739, −6.49197604262756310956098737221, −5.88653772551027016389549044951, −5.35605791333955760175824432625, −4.13395860372219760280279058959, −3.30702900675108618496780519866, −2.09547899425946937071388341247, −0.917225692207709623533283917061, 1.28703322059709379227229484600, 2.20809680650128159447433353229, 2.90996562213687026031525663194, 4.55647441287209308633439098334, 5.42523657276802890413332171905, 6.09475373103717147297282332110, 6.54950712232408697235156165199, 7.47106461984342878078020704486, 8.566040852994502541309657990025, 9.396660362348267953784664868436

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