↑

Properties

Label 2-2028-13.10-c1-0-16
Degree $2$
Conductor $2028$
Sign $0.425 + 0.905i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + 1.69i·5-s + (0.266 + 0.153i)7-s + (−0.499 + 0.866i)9-s + (−0.652 + 0.376i)11-s + (1.46 − 0.846i)15-s + (2.02 − 3.50i)17-s + (−6.19 − 3.57i)19-s − 0.307i·21-s + (−2.95 − 5.11i)23-s + 2.13·25-s + 0.999·27-s + (0.480 + 0.832i)29-s + 4.02i·31-s + (0.652 + 0.376i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.756i·5-s + (0.100 + 0.0582i)7-s + (−0.166 + 0.288i)9-s + (−0.196 + 0.113i)11-s + (0.378 − 0.218i)15-s + (0.491 − 0.850i)17-s + (−1.42 − 0.821i)19-s − 0.0672i·21-s + (−0.616 − 1.06i)23-s + 0.427·25-s + 0.192·27-s + (0.0892 + 0.154i)29-s + 0.723i·31-s + (0.113 + 0.0655i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.425 + 0.905i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ 0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.286844852\)
\(L(\frac12)\) \(\approx\) \(1.286844852\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69iT - 5T^{2} \)
7 \( 1 + (-0.266 - 0.153i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.652 - 0.376i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.02 + 3.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.19 + 3.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.95 + 5.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.480 - 0.832i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.02iT - 31T^{2} \)
37 \( 1 + (-3.25 + 1.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.73 + 2.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.10 + 3.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.03iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (-11.7 - 6.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.62 + 8.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.9 + 6.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.79 + 1.03i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.52iT - 73T^{2} \)
79 \( 1 + 8.29T + 79T^{2} \)
83 \( 1 + 8.97iT - 83T^{2} \)
89 \( 1 + (-7.99 + 4.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.5 + 7.79i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.850863588956986434484974741919, −8.225959218634678026538224224592, −7.18389615756641867858547313924, −6.81049178103028648268371265298, −5.95776404219534694764359394075, −5.03332074357323417915398230121, −4.11582179786286009336689523289, −2.83663937930421235057776183408, −2.18065967684037719632093182373, −0.55146817119029563305627872332, 1.08739797730589218400070405715, 2.36093930820499715876349194288, 3.82134012243762858912369538320, 4.25644907735466638080865050423, 5.39276193824151740122944576256, 5.87154257662494702909408582726, 6.84031238499377749965781938872, 8.099040434094945423131674274022, 8.331477601544947771587467236932, 9.372519963537914537189953667298

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