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Properties

Label 1-52-52.31-r0-0-0
Degree $1$
Conductor $52$
Sign $0.289 - 0.957i$
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  − 3-s − i·5-s − i·7-s + 9-s − i·11-s + i·15-s − 17-s + i·19-s + i·21-s + 23-s − 25-s − 27-s + 29-s + i·31-s + i·33-s + ⋯
L(s)  = 1  − 3-s − i·5-s − i·7-s + 9-s − i·11-s + i·15-s − 17-s + i·19-s + i·21-s + 23-s − 25-s − 27-s + 29-s + i·31-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(0.241486\)
Root analytic conductor: \(0.241486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 52,\ (0:\ ),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5033173096 - 0.3734896139i\)
\(L(\frac12)\) \(\approx\) \(0.5033173096 - 0.3734896139i\)
\(L(1)\) \(\approx\) \(0.7157601546 - 0.2503442995i\)
\(L(1)\) \(\approx\) \(0.7157601546 - 0.2503442995i\)

Euler product

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

−33.811098587390012161795304273594, −32.92191208698429055071187161718, −31.17737029832455341324780052493, −30.31542162811635099575021981413, −29.00229578408942354103183746248, −28.16182780514288828767367685231, −26.99953108037411679421030764739, −25.703916255785621391722325201917, −24.426260635854402500225042643267, −23.03059733238106065017315218596, −22.27973293448546190858383162662, −21.31976357707914301391926078331, −19.40462303564601644646425107951, −18.17425756703542261728447595996, −17.52224141715457118683231417200, −15.73604014685437549145276867868, −14.947665516908693707124256508405, −13.042531542370043693581191092322, −11.75670724151034048905801251293, −10.7701869326978400234264166023, −9.35767548911718519646635573235, −7.23159114549101019373834421258, −6.15978730408381988654908335204, −4.640535876840447962826454937334, −2.42206321115837043014575335913, 1.03855202346103598078575020783, 4.07396783475317205072826012576, 5.35699686341634239458041907286, 6.83168493688293992638838880850, 8.51183994585278096512360741982, 10.200815103974160151747549203593, 11.362660144913273090735013107041, 12.691435314523025139420155035766, 13.76759567466002012390614326394, 15.86186627230425363820986821410, 16.72703652485657162139907040153, 17.57941766952002539763570152377, 19.171117199400549206948670200505, 20.54428061797321589975637400113, 21.63271063182726164916860186874, 23.037720836862041473159974388438, 23.914021253384195962872171717245, 24.8590560711499689881313598121, 26.78329039729185853220037753732, 27.50405089756677712528147837792, 28.99577057186073347048304725178, 29.345218281904380607806380559328, 30.9306955094768431766537413031, 32.47813064747680807844138012846, 33.10193252347745468394589453081

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