Properties

Degree 1
Conductor 53
Sign $0.327 + 0.944i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.120 − 0.992i)2-s + (−0.885 + 0.464i)3-s + (−0.970 + 0.239i)4-s + (−0.568 + 0.822i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (0.354 + 0.935i)8-s + (0.568 − 0.822i)9-s + (0.885 + 0.464i)10-s + (−0.748 + 0.663i)11-s + (0.748 − 0.663i)12-s + (−0.970 − 0.239i)13-s + (0.970 − 0.239i)14-s + (0.120 − 0.992i)15-s + (0.885 − 0.464i)16-s + (−0.354 + 0.935i)17-s + ⋯
L(s,χ)  = 1  + (−0.120 − 0.992i)2-s + (−0.885 + 0.464i)3-s + (−0.970 + 0.239i)4-s + (−0.568 + 0.822i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (0.354 + 0.935i)8-s + (0.568 − 0.822i)9-s + (0.885 + 0.464i)10-s + (−0.748 + 0.663i)11-s + (0.748 − 0.663i)12-s + (−0.970 − 0.239i)13-s + (0.970 − 0.239i)14-s + (0.120 − 0.992i)15-s + (0.885 − 0.464i)16-s + (−0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.327 + 0.944i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.327 + 0.944i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $0.327 + 0.944i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (38, \cdot )$
Sato-Tate  :  $\mu(26)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (0:\ ),\ 0.327 + 0.944i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3204473951 + 0.2280575937i$
$L(\frac12,\chi)$  $\approx$  $0.3204473951 + 0.2280575937i$
$L(\chi,1)$  $\approx$  0.5477725599 + 0.05246738199i
$L(1,\chi)$  $\approx$  0.5477725599 + 0.05246738199i

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

−33.41121127645312489042257884153, −32.18234705725964746047364607357, −31.15044762544620838968426775133, −29.55400701999535489941030299628, −28.46710689642330686484530447175, −27.288114105542197414883643703126, −26.479684710991426986297393177468, −24.57329901487825361340469519828, −24.08638358620307429084147440000, −23.19923855509454849837321030977, −22.03631510047771685133809382925, −20.144106350591616608087512432632, −18.76753918875284315868443540417, −17.54780547141830208132269663164, −16.55334895855837144827102233326, −15.86891612206029468742912113707, −13.9163067438548154658659814903, −12.91941908800691813301502152368, −11.43730465052852659833968529737, −9.78660511892688359933296547707, −7.950526483430512118264563116665, −7.15630709519020729840705975928, −5.45010711987083356259513273487, −4.42222006617029809938976647002, −0.612159535860909664861587249779, 2.52459785666465709947536817681, 4.22490615204732946348089899084, 5.66067712013181414553316947980, 7.74291352105032867696475582030, 9.62669774092673959990554065844, 10.62375580407183962229660946819, 11.78723156704589705549158216746, 12.56129101987428518152412620337, 14.671199535243020061585096741374, 15.753423599649223413195215646273, 17.60299101802147093268056763766, 18.30705918241720131674250524261, 19.54215997292413721875826320984, 21.03025956605165773022785691019, 22.137025317112986863410288456638, 22.67923472197273187396284352982, 24.00982599865654536234302241191, 26.142970117806649000892087955870, 27.047612880039158092706967693894, 28.11696920845610788680363342167, 28.82332940917485892032720781978, 30.117095785658296225151994820267, 31.14399421505179446097826835804, 32.17311970749340295027667842766, 33.828930354086180800728197465309

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