Properties

Label 1-52-52.19-r0-0-0
Degree $1$
Conductor $52$
Sign $0.884 + 0.466i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(0.241486\)
Root analytic conductor: \(0.241486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 52,\ (0:\ ),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9730201566 + 0.2406863629i\)
\(L(\frac12)\) \(\approx\) \(0.9730201566 + 0.2406863629i\)
\(L(1)\) \(\approx\) \(1.115542137 + 0.1926393906i\)
\(L(1)\) \(\approx\) \(1.115542137 + 0.1926393906i\)

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

−33.68252336536203620806692948423, −31.9832973936763093747543366608, −30.90550810013843931871090231839, −30.06530802642846333267329106032, −29.280726852082216169272719725220, −27.507025392064643183606484573803, −26.30231923239819669484577191723, −25.530657732546820449500331370699, −23.94749273056359017369558586828, −23.409973257568655736407087037848, −21.687905082227729903282352851367, −20.48577630516190729637487005718, −19.09802827429505360552467942147, −18.31377542429120413444888674120, −17.15539069778570406554796692127, −15.12077324070295971124778728460, −14.22943483290959525382224468610, −13.1286616294813146035455421259, −11.51458378740287426414035550289, −10.31977773516113258815759993293, −8.27402548102020332203683418405, −7.40639648583334680534506335241, −5.92109750156327252711191868739, −3.575537049712194291785502663057, −1.96277225190173613044656552891, 2.33993580444931292873855472596, 4.44441703944746924986139708699, 5.32653210231909546308575715373, 7.92947442382310076582250579505, 8.87640419662040725682587795929, 10.18609695282364257068897489772, 11.713471581629377526371759732499, 13.196069879659542207295953049358, 14.641823126569829591916960772783, 15.69897273316911315954608072928, 16.80010213507133469175813663063, 18.23172873602840391860350065980, 19.91386934854839961278518131794, 20.863857142904898660331951440515, 21.5308760340493286533156075710, 23.19872760524660128053501741625, 24.55307985464701800341712220842, 25.49341386683074255505170454282, 26.83545300236801771562270749512, 27.92111292178228931259658935422, 28.56323226098773587679925198532, 30.44775188966919322294088010740, 31.59354152304034093698577513673, 32.198874478200424875395174707885, 33.535369857879609972592661680418

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