↑

Properties

Label 2-2028-156.83-c0-0-2
Degree $2$
Conductor $2028$
Sign $0.289 - 0.957i$
Analytic cond. $1.01210$
Root an. cond. $1.00603$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + i·3-s + 4-s + (1 + i)5-s + i·6-s + 8-s − 9-s + (1 + i)10-s + (−1 − i)11-s + i·12-s + (−1 + i)15-s + 16-s − 18-s + (1 + i)20-s + (−1 − i)22-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + (1 + i)5-s + i·6-s + 8-s − 9-s + (1 + i)10-s + (−1 − i)11-s + i·12-s + (−1 + i)15-s + 16-s − 18-s + (1 + i)20-s + (−1 − i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(1.01210\)
Root analytic conductor: \(1.00603\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :0),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.412248121\)
\(L(\frac12)\) \(\approx\) \(2.412248121\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
13 \( 1 \)
good5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1 + i)T + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (1 - i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (-1 + i)T - iT^{2} \)
97 \( 1 - iT^{2} \)
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

−9.876879703047140033974836932697, −8.752420370050212576768357792820, −7.86538740067148481780510366832, −6.81551346389006191772691731010, −6.03083130797490311461949899858, −5.51576028221521561612866618652, −4.75703565005104990374698086678, −3.54281169703380792057622594744, −2.98500097985057491188391927604, −2.13934075652067339772505090270, 1.47045258867578503534983092260, 2.15735424167193199639045689658, 3.12508719468420992256680288802, 4.65742247256310147758026385416, 5.14143557997279458546594539263, 5.93366591095947903656859641897, 6.61796763066738775638114828300, 7.53244087918439376241624192251, 8.129168469422577155169664827657, 9.138514341803799439749752079966

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