Properties

Degree 1
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Sign $-0.850 - 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $-0.850 - 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{1020} (407, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1020,\ (1:\ ),\ -0.850 - 0.525i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.03370639731 - 0.1186514741i$
$L(\frac12,\chi)$  $\approx$  $0.03370639731 - 0.1186514741i$
$L(\chi,1)$  $\approx$  0.8762668408 + 0.07901293174i
$L(1,\chi)$  $\approx$  0.8762668408 + 0.07901293174i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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