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Properties

Label 2-50-5.2-c2-0-2
Degree $2$
Conductor $50$
Sign $0.229 + 0.973i$
Analytic cond. $1.36240$
Root an. cond. $1.16721$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (3 − 3i)3-s + 2i·4-s − 6·6-s + (−3 − 3i)7-s + (2 − 2i)8-s − 9i·9-s + 12·11-s + (6 + 6i)12-s + (−12 + 12i)13-s + 6i·14-s − 4·16-s + (12 + 12i)17-s + (−9 + 9i)18-s + 20i·19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1 − i)3-s + 0.5i·4-s − 6-s + (−0.428 − 0.428i)7-s + (0.250 − 0.250i)8-s − i·9-s + 1.09·11-s + (0.5 + 0.5i)12-s + (−0.923 + 0.923i)13-s + 0.428i·14-s − 0.250·16-s + (0.705 + 0.705i)17-s + (−0.5 + 0.5i)18-s + 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(1.36240\)
Root analytic conductor: \(1.16721\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.900646 - 0.712788i\)
\(L(\frac12)\) \(\approx\) \(0.900646 - 0.712788i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 \)
good3 \( 1 + (-3 + 3i)T - 9iT^{2} \)
7 \( 1 + (3 + 3i)T + 49iT^{2} \)
11 \( 1 - 12T + 121T^{2} \)
13 \( 1 + (12 - 12i)T - 169iT^{2} \)
17 \( 1 + (-12 - 12i)T + 289iT^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + (-3 + 3i)T - 529iT^{2} \)
29 \( 1 + 30iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 + (48 + 48i)T + 1.36e3iT^{2} \)
41 \( 1 + 48T + 1.68e3T^{2} \)
43 \( 1 + (27 - 27i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27 - 27i)T + 2.20e3iT^{2} \)
53 \( 1 + (12 - 12i)T - 2.80e3iT^{2} \)
59 \( 1 + 60iT - 3.48e3T^{2} \)
61 \( 1 - 32T + 3.72e3T^{2} \)
67 \( 1 + (3 + 3i)T + 4.48e3iT^{2} \)
71 \( 1 + 48T + 5.04e3T^{2} \)
73 \( 1 + (12 - 12i)T - 5.32e3iT^{2} \)
79 \( 1 - 40iT - 6.24e3T^{2} \)
83 \( 1 + (-93 + 93i)T - 6.88e3iT^{2} \)
89 \( 1 - 30iT - 7.92e3T^{2} \)
97 \( 1 + (-12 - 12i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.64661998336004243774261294396, −13.92663883420789206712279376832, −12.69076772712617704935233109361, −11.86245207177202507846598702559, −10.08745848297075308675851995583, −8.987771474630974893119655693889, −7.77591223426661159754654466015, −6.69598080613287623425340706945, −3.67919922508282971486074977192, −1.80025676486837353071318586815, 3.15730387915536506833046377401, 5.08200756604778021156898137371, 7.02697502479373260278127599454, 8.597359020093749534253980411125, 9.411306383803152646954322466506, 10.29476229216911077361313726089, 12.04540692977967871954181129679, 13.75151866857549705537000314050, 14.82993393552372711007767059544, 15.39375256660699070522923994612

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