↑

Properties

Label 1-2028-2028.947-r1-0-0
Degree $1$
Conductor $2028$
Sign $-0.301 + 0.953i$
Analytic cond. $217.938$
Root an. cond. $217.938$
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  + (0.663 + 0.748i)5-s + (−0.721 + 0.692i)7-s + (0.903 − 0.428i)11-s + (0.692 + 0.721i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s + (−0.120 + 0.992i)25-s + (0.996 − 0.0804i)29-s + (0.992 − 0.120i)31-s + (−0.996 − 0.0804i)35-s + (−0.391 + 0.919i)37-s + (0.534 + 0.845i)41-s + (−0.919 + 0.391i)43-s + (−0.935 + 0.354i)47-s + (0.0402 − 0.999i)49-s + ⋯
L(s)  = 1  + (0.663 + 0.748i)5-s + (−0.721 + 0.692i)7-s + (0.903 − 0.428i)11-s + (0.692 + 0.721i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s + (−0.120 + 0.992i)25-s + (0.996 − 0.0804i)29-s + (0.992 − 0.120i)31-s + (−0.996 − 0.0804i)35-s + (−0.391 + 0.919i)37-s + (0.534 + 0.845i)41-s + (−0.919 + 0.391i)43-s + (−0.935 + 0.354i)47-s + (0.0402 − 0.999i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.301 + 0.953i$
Analytic conductor: \(217.938\)
Root analytic conductor: \(217.938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (1:\ ),\ -0.301 + 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448977981 + 1.978833844i\)
\(L(\frac12)\) \(\approx\) \(1.448977981 + 1.978833844i\)
\(L(1)\) \(\approx\) \(1.165668472 + 0.3825803396i\)
\(L(1)\) \(\approx\) \(1.165668472 + 0.3825803396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.663 + 0.748i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (0.903 - 0.428i)T \)
17 \( 1 + (0.692 + 0.721i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.996 - 0.0804i)T \)
31 \( 1 + (0.992 - 0.120i)T \)
37 \( 1 + (-0.391 + 0.919i)T \)
41 \( 1 + (0.534 + 0.845i)T \)
43 \( 1 + (-0.919 + 0.391i)T \)
47 \( 1 + (-0.935 + 0.354i)T \)
53 \( 1 + (0.970 - 0.239i)T \)
59 \( 1 + (0.316 - 0.948i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (0.160 - 0.987i)T \)
71 \( 1 + (0.999 - 0.0402i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (0.354 + 0.935i)T \)
83 \( 1 + (0.464 + 0.885i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.979 - 0.200i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.606361674188527879275914093962, −19.00472614554814146146125254196, −17.746438788747291799260237906686, −17.37364841919441840512808815552, −16.66160180210366167435995182387, −16.07103380956338213344575495927, −15.19049323023286466442081567138, −14.16694748614470423014365815666, −13.69395729329677552756092826010, −12.925862014620845611551703376319, −12.26943791351322858896068209676, −11.50647672018473833292490155201, −10.35625768782058953228262478734, −9.85331079704592223910952668416, −9.13114155083213913621539312035, −8.467970065219381071197913874756, −7.25729210637225062106029298248, −6.719440658282371650318323750314, −5.85245163957422987058686658473, −4.94120057207187894780117053614, −4.20018919178303396118566197354, −3.29620200116353532540648616906, −2.235898470114724559326693896459, −1.19559762388778534278734452133, −0.48952820844346733630816938681, 0.99505432135176060843481935508, 2.02435820260930944024448670661, 2.92298876994804984155158168745, 3.53627488526229233607252880746, 4.6650475683144853404953080422, 5.771042535595970923026945839450, 6.45169074420331767492250499953, 6.6826164563333573908480693767, 8.160060765744889221269922960151, 8.692424410899845365522681512858, 9.75652658908105411026932058092, 10.11516062161473784184917479333, 11.04185963113143009259049408873, 11.89507736947015086889332761265, 12.60995633722929653980171689348, 13.36185644672726882163186900584, 14.224512799758467404148448497442, 14.80418472823356868496065501071, 15.41205443451435938342601205454, 16.56915260885091658234165183691, 16.90498312040194534712465365681, 17.86604160931868924058647024385, 18.5882283526497159009327505470, 19.24860815622873688987092834305, 19.601807743701854943143082926331

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