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Properties

Label 2-2028-13.3-c1-0-22
Degree $2$
Conductor $2028$
Sign $-0.477 + 0.878i$
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 − 1.73i)7-s + (−0.499 + 0.866i)9-s + (3 − 5.19i)17-s + (1 − 1.73i)19-s − 1.99·21-s − 5·25-s + 0.999·27-s + (3 + 5.19i)29-s − 2·31-s + (1 + 1.73i)37-s + (−6 − 10.3i)41-s + (2 − 3.46i)43-s + (1.50 + 2.59i)49-s − 6·51-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.377 − 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.727 − 1.26i)17-s + (0.229 − 0.397i)19-s − 0.436·21-s − 25-s + 0.192·27-s + (0.557 + 0.964i)29-s − 0.359·31-s + (0.164 + 0.284i)37-s + (−0.937 − 1.62i)41-s + (0.304 − 0.528i)43-s + (0.214 + 0.371i)49-s − 0.840·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.477 + 0.878i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (2005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322161762\)
\(L(\frac12)\) \(\approx\) \(1.322161762\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.855392949908671382398169944737, −7.922470784541478746940733653123, −7.28806576741047158165687735811, −6.74148830585652666458761979340, −5.58300596188313059253934211348, −5.01017830832325605017387808174, −3.94175298666595464156501267267, −2.90210210365423575400946996874, −1.68059838730049651040167192149, −0.51412935942636217901115293525, 1.39948204759831846211740226576, 2.61256076316698456969614148543, 3.75131768004363465432950239100, 4.47634040036931869886842646176, 5.66417264987268362089296624161, 5.86099930707404484695664078826, 7.02789847919226127353579598533, 8.125264509234280050430023591540, 8.452159378971536963249481942154, 9.578678100228117679164423842465

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