Properties

Degree 1
Conductor $ 2^{2} \cdot 13 $
Sign $-0.872 - 0.488i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)3-s − 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s − 27-s + (−0.5 + 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)3-s − 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s − 27-s + (−0.5 + 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.872 - 0.488i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.872 - 0.488i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(52\)    =    \(2^{2} \cdot 13\)
\( \varepsilon \)  =  $-0.872 - 0.488i$
motivic weight  =  \(0\)
character  :  $\chi_{52} (23, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 52,\ (1:\ ),\ -0.872 - 0.488i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2331095218 - 0.8928365098i$
$L(\frac12,\chi)$  $\approx$  $0.2331095218 - 0.8928365098i$
$L(\chi,1)$  $\approx$  0.7489374876 - 0.4453009972i
$L(1,\chi)$  $\approx$  0.7489374876 - 0.4453009972i

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

−33.67293673466283136575945359220, −32.10194079608307526453355591017, −31.71582827299190179796649799935, −30.618140311887031979488397161626, −28.86256242067640018309477291507, −27.80047936977226205495883992922, −26.84817782481030006899895881102, −25.859376625123765100625167960985, −24.568400587829136468191818310, −23.139773838616662421060747483144, −21.95430530112894974300839969570, −20.96088708450845907096000102266, −19.52949609315111347571800825820, −18.84991074966535044148330586240, −16.744418837291928501890046569989, −15.63997907802044154233509387413, −14.99696501081263347052604337304, −13.292762590665398899677856169, −11.73685896141873834825034309878, −10.47617913204730531525878394447, −8.929434658819177238132818911401, −7.98536572701750626561173370992, −5.82367572759192724853854517167, −4.1204496298414977802048061276, −2.85996825068437282631126703886, 0.48614862184044773147667988502, 2.75958767291290139367481725149, 4.388888380289097892188239982809, 6.83625055426313733961191276370, 7.57133020025015155670851824802, 9.059951294348558988831548255129, 10.86282576597423750337601518927, 12.355784484830705618601164162938, 13.30222370085365280038006603741, 14.7147151006330032927465202943, 15.9615674953239894473787771758, 17.51322739124647853466441198870, 18.79823869674210963747076080809, 19.851086194850863417343038108245, 20.54517686192423355565595791586, 22.76046840473330069991586894729, 23.45275916012093043079372044715, 24.550415951237547923113758727197, 25.93932912475006906005037151823, 26.731169794879473922442152768303, 28.26686356275220088147916738544, 29.49960624060977585547451880135, 30.56394468655111312957162213550, 31.3752001694123878548435491825, 32.481024431824059343273458810675

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