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Properties

Label 1-2028-2028.227-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

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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: $\chi_{2028} (227, \cdot )$
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 \( 1 + (-0.979 + 0.200i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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