Properties

Degree 1
Conductor $ 5^{2} \cdot 7 \cdot 23 $
Sign $0.520 + 0.853i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.113 − 0.993i)2-s + (0.336 − 0.941i)3-s + (−0.974 + 0.226i)4-s + (−0.974 − 0.226i)6-s + (0.336 + 0.941i)8-s + (−0.774 − 0.633i)9-s + (−0.736 − 0.676i)11-s + (−0.113 + 0.993i)12-s + (0.170 + 0.985i)13-s + (0.897 − 0.441i)16-s + (−0.226 + 0.974i)17-s + (−0.540 + 0.841i)18-s + (0.0855 − 0.996i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s,χ)  = 1  + (−0.113 − 0.993i)2-s + (0.336 − 0.941i)3-s + (−0.974 + 0.226i)4-s + (−0.974 − 0.226i)6-s + (0.336 + 0.941i)8-s + (−0.774 − 0.633i)9-s + (−0.736 − 0.676i)11-s + (−0.113 + 0.993i)12-s + (0.170 + 0.985i)13-s + (0.897 − 0.441i)16-s + (−0.226 + 0.974i)17-s + (−0.540 + 0.841i)18-s + (0.0855 − 0.996i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.520 + 0.853i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.520 + 0.853i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $0.520 + 0.853i$
motivic weight  =  \(0\)
character  :  $\chi_{4025} (48, \cdot )$
Sato-Tate  :  $\mu(220)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4025,\ (0:\ ),\ 0.520 + 0.853i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1808257440 - 0.1014964298i$
$L(\frac12,\chi)$  $\approx$  $-0.1808257440 - 0.1014964298i$
$L(\chi,1)$  $\approx$  0.5270699557 - 0.5859223587i
$L(1,\chi)$  $\approx$  0.5270699557 - 0.5859223587i

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

−18.73710074638083719064013590942, −18.2603407340049503048553097953, −17.60422048192287463068614133958, −16.788226200091863194859689621488, −16.16364446536599883222470225828, −15.740058205398227793295791356598, −14.96993975911945982610895069655, −14.60026435895078549071997516474, −13.80609328405904448500863910506, −13.065676770810626637642897379253, −12.45963881224049531768334245903, −11.23463888040035701770653910700, −10.53260115455414691045887926831, −9.84235682395337688398997612036, −9.40020026093651712836364805194, −8.54516896106649275684851628585, −7.80702374965876063218832135411, −7.451326823124131684891893934415, −6.260949352802934685728336277860, −5.5601835715080814929629374976, −4.91597675397456365321440346631, −4.36310623240114617919393178713, −3.36466091702543818315112714738, −2.73243091685107433353451630827, −1.38637937938939502417767240510, 0.06446574148368448120312973436, 1.00499927526799555064708539433, 2.073348038108857639366630363914, 2.352405021046555258884376882603, 3.44114306205248531941235556957, 3.98073888689836627914866397532, 5.08500013033043942676546185358, 5.856501525755375816803334230053, 6.70977237386405096578052253914, 7.60445852259070435734275511096, 8.2401985731508680267467531947, 8.96949933323833218176617562953, 9.37210352812462866263373599989, 10.57982978977971609211108851888, 11.04454426964068539832426336103, 11.75095930360426799280759377011, 12.4335027608575989894554793273, 13.132633809015209764007657954622, 13.57751396971393495935215621662, 14.17034955668063522638736994331, 14.95345394235755028849718109360, 15.87973891295900127803628759299, 16.90710151575370914489488057241, 17.39197749488459173988075651896, 18.2048971059287318005949902766

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