Properties

Degree 1
Conductor $ 2^{3} \cdot 13 $
Sign $-0.957 - 0.289i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s + i·5-s + i·7-s + 9-s i·11-s i·15-s − 17-s + i·19-s i·21-s − 23-s − 25-s − 27-s − 29-s i·31-s + i·33-s + ⋯
L(s,χ)  = 1  − 3-s + i·5-s + i·7-s + 9-s i·11-s i·15-s − 17-s + i·19-s i·21-s − 23-s − 25-s − 27-s − 29-s i·31-s + i·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.957 - 0.289i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.957 - 0.289i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(104\)    =    \(2^{3} \cdot 13\)
\( \varepsilon \)  =  $-0.957 - 0.289i$
motivic weight  =  \(0\)
character  :  $\chi_{104} (5, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 104,\ (1:\ ),\ -0.957 - 0.289i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.03549308692 + 0.2397068222i$
$L(\frac12,\chi)$  $\approx$  $-0.03549308692 + 0.2397068222i$
$L(\chi,1)$  $\approx$  0.5896806770 + 0.1785409430i
$L(1,\chi)$  $\approx$  0.5896806770 + 0.1785409430i

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

−28.74773727419156759245152690341, −28.12671862199252356156213723112, −27.12126330723912518095216256273, −25.92039501628253229625300199782, −24.446055307379791977571898482055, −23.78987785580122184807518046961, −22.84887163546698866287452069959, −21.758947770471268121026484517984, −20.48313951514751819791611690192, −19.75698175345698899749115838607, −17.97012514052447120862961398462, −17.299105567099020438728504314346, −16.37886346450820465072081248230, −15.36988401689207659325177850398, −13.55101827183204449210462405576, −12.711375058735052779432976724393, −11.59911162948404428191720909436, −10.41960509504470338642998829976, −9.30350124558988017840851960786, −7.628570257591711888304802724853, −6.50797872571950845249149955906, −4.946625639809738240268094188345, −4.224102755157359037302939845538, −1.58277320383327534310472232191, −0.118225366233975235185216207309, 2.16408054942311863700230737045, 3.86041191888646508791778259121, 5.68198137517492393137444780698, 6.293479942169842924520633483766, 7.78006975434942007726717894021, 9.4119398685242990447046346503, 10.770843224219191015580802842227, 11.47952369454014867724831207300, 12.62178936255578844442337313251, 14.06438290959327229607127453354, 15.35064383048127301623231057123, 16.21103371935390479088763262190, 17.57610256422204852576705240079, 18.49022554263852922285608734677, 19.13521263686051600045093390449, 21.02930947074692448592870926761, 22.16327900189866611507259194817, 22.4253448145791674100649335163, 23.865192902755416495838185301802, 24.71980946370176285671740514301, 26.105397810164068015539884279575, 27.10378292513532574349387336555, 28.03006693351008376914981285162, 29.14091306439309205881993268738, 29.818799709033310369186178057784

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