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Properties

Label 2-2028-13.3-c1-0-26
Degree $2$
Conductor $2028$
Sign $-0.872 + 0.488i$
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 + 2·5-s + (2 − 3.46i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + (−1 − 1.73i)15-s + (−1 + 1.73i)17-s − 3.99·21-s + (−4 − 6.92i)23-s − 25-s + 0.999·27-s + (−1 − 1.73i)29-s − 8·31-s + (−3 + 5.19i)33-s + (4 − 6.92i)35-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.894·5-s + (0.755 − 1.30i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + (−0.258 − 0.447i)15-s + (−0.242 + 0.420i)17-s − 0.872·21-s + (−0.834 − 1.44i)23-s − 0.200·25-s + 0.192·27-s + (−0.185 − 0.321i)29-s − 1.43·31-s + (−0.522 + 0.904i)33-s + (0.676 − 1.17i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.872 + 0.488i$
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: \((2,\ 2028,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387633883\)
\(L(\frac12)\) \(\approx\) \(1.387633883\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.548583308449223808370208091842, −8.066296238271871124932920325157, −7.30985316203668655454752940652, −6.28159525747618395074265400778, −5.81530488679215928429099206513, −4.85837981214655779753889615028, −3.91061370836501476058584103792, −2.67699246988239394020151710013, −1.60758720794686890538022820033, −0.47905421813076879068820669446, 1.97650801983264255256554280590, 2.25821418922262888806217495572, 3.78835736438860814719252569264, 4.95857582726280406467229117540, 5.39616740895194054846133420308, 5.96408806509461382819148215494, 7.23961289993826900816749729561, 7.84364284147747870854718631821, 9.071150834859523920568686744617, 9.339369091268400446961716809545

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