Properties

Label 1-52-52.15-r0-0-0
Degree $1$
Conductor $52$
Sign $0.477 - 0.878i$
Analytic cond. $0.241486$
Root an. cond. $0.241486$
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 i·5-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·21-s + (−0.5 + 0.866i)23-s − 25-s − 27-s + (−0.5 + 0.866i)29-s i·31-s + (0.866 − 0.5i)33-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s i·5-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·21-s + (−0.5 + 0.866i)23-s − 25-s − 27-s + (−0.5 + 0.866i)29-s i·31-s + (0.866 − 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(0.241486\)
Root analytic conductor: \(0.241486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 52,\ (0:\ ),\ 0.477 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8113134835 - 0.4824666976i\)
\(L(\frac12)\) \(\approx\) \(0.8113134835 - 0.4824666976i\)
\(L(1)\) \(\approx\) \(1.008432410 - 0.3743251013i\)
\(L(1)\) \(\approx\) \(1.008432410 - 0.3743251013i\)

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 \( 1 - iT \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−33.4612076135661110716322163537, −32.598112542238019905809802899992, −31.546601904742011694915609326631, −30.272912657030889623951250997479, −29.27129032025964943269196890358, −27.6693708030242225991026658229, −26.65459561868199966084668066735, −25.9801985375403338083760459633, −24.74207554536101707691268189942, −22.68485433299358497072920095247, −22.36714500205458038703503728994, −20.855447554307605177437142518361, −19.669457907846057272051578894564, −18.665482802413952056756565210259, −16.84455902307985961113052664605, −15.85100221675302379412875489627, −14.48911727298974742556966853600, −13.69846051657300168303630676056, −11.60865316285847333600719433102, −10.27988488578117742584337442162, −9.387847552481165577054408399814, −7.59196732415173634910989196993, −6.06661818175461017362427215795, −3.96713462095470882962957348290, −2.911774947049750184837037758224, 1.594344909186465607486496135145, 3.543778027803707331971337932293, 5.65865003222743779439604456608, 7.118871054388509400601730555779, 8.65479939217735953144787239716, 9.56385130074579473761258416431, 11.99075933269504376238149852323, 12.68016620188670958722716708569, 13.89275246709670516198497570793, 15.39301724092766888996275662592, 16.798358020688816216639308250243, 18.03781345044416577382605048980, 19.50832942047579863056275947718, 20.03465887703240906528573038656, 21.601805436819209155834469841313, 23.092798173623892469495352099024, 24.27684419558324045336901634080, 25.16634257370934945713832025785, 26.03883110551415798928535961649, 27.84852352326918570443574975224, 28.77087060674976115076006935860, 29.88244729957315520371571420395, 31.09865760863963456472685008357, 32.06074664895349836424746949139, 32.93283785916200077966705338440

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