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Properties

Label 1-1020-1020.203-r1-0-0
Degree $1$
Conductor $1020$
Sign $-0.850 + 0.525i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  − i·7-s − 11-s + i·13-s + 19-s − i·23-s − 29-s + 31-s + i·37-s + 41-s − i·43-s − i·47-s − 49-s − i·53-s − 59-s − 61-s + ⋯
L(s)  = 1  − i·7-s − 11-s + i·13-s + 19-s − i·23-s − 29-s + 31-s + i·37-s + 41-s − i·43-s − i·47-s − 49-s − i·53-s − 59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03370639731 + 0.1186514741i\)
\(L(\frac12)\) \(\approx\) \(0.03370639731 + 0.1186514741i\)
\(L(1)\) \(\approx\) \(0.8762668408 - 0.07901293174i\)
\(L(1)\) \(\approx\) \(0.8762668408 - 0.07901293174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - iT \)
97 \( 1 \)
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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

−21.15383408466093308156952164623, −20.36506234021942362972860644694, −19.51334620612594674059909430137, −18.63701538799572224074545033708, −18.018210113459069614614370346955, −17.389898183796880591194302046503, −16.09906475611141316912702709014, −15.63446686898388133555367539320, −14.960519468388983681229495584192, −13.93115836022582031282704838198, −13.03567887710446278783242529220, −12.433477676654488585305465188696, −11.49147373120826250900587647961, −10.71696828605389383990607568201, −9.71717690085353621224893019244, −9.05591269585805605146746661873, −7.89635926808293584209150858340, −7.520981030696059877537707481437, −5.96844283045978770557109274956, −5.56602332228144094029825940742, −4.61310240732977421154928055708, −3.19921714888382086478148262472, −2.64752570628475380841729316720, −1.39107674823859612055520669447, −0.02722066220917657539256894249, 1.09416969858867942658062225610, 2.27595109148774309125493958719, 3.349187265836932139192320669087, 4.33663347740049202357336110941, 5.090933950847478460359713644123, 6.24319149951510606003938107342, 7.14259328210094764390039955845, 7.80084975503973060100116074694, 8.790533530090761653504669100433, 9.81628750630894019447223195106, 10.458257438407715634282286943320, 11.30906495796690017115156517039, 12.14299395585258436156587219442, 13.21449448969374285972205010693, 13.7362536131058654810018041606, 14.530015090506306356689886754584, 15.52373810470186134499015347968, 16.37029409830392379318978611046, 16.88857896175714051367408175951, 17.87728977626029466885205954675, 18.62457032467522282732429246898, 19.34336765918738300116701301856, 20.44193832090846375725059972832, 20.70667071978066302046110070260, 21.695729722186123722217144840787

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