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Properties

Label 1-52-52.23-r1-0-0
Degree $1$
Conductor $52$
Sign $-0.872 - 0.488i$
Analytic cond. $5.58817$
Root an. cond. $5.58817$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(5.58817\)
Root analytic conductor: \(5.58817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 52,\ (1:\ ),\ -0.872 - 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2331095218 - 0.8928365098i\)
\(L(\frac12)\) \(\approx\) \(0.2331095218 - 0.8928365098i\)
\(L(1)\) \(\approx\) \(0.7489374876 - 0.4453009972i\)
\(L(1)\) \(\approx\) \(0.7489374876 - 0.4453009972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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