Properties

Label 2-50-5.4-c21-0-4
Degree $2$
Conductor $50$
Sign $-0.894 - 0.447i$
Analytic cond. $139.738$
Root an. cond. $11.8211$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.02e3i·2-s + 9.45e4i·3-s − 1.04e6·4-s + 9.67e7·6-s + 3.65e8i·7-s + 1.07e9i·8-s + 1.52e9·9-s − 3.92e10·11-s − 9.90e10i·12-s + 4.64e11i·13-s + 3.74e11·14-s + 1.09e12·16-s + 2.43e11i·17-s − 1.56e12i·18-s + 6.74e12·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.924i·3-s − 0.5·4-s + 0.653·6-s + 0.488i·7-s + 0.353i·8-s + 0.146·9-s − 0.455·11-s − 0.462i·12-s + 0.934i·13-s + 0.345·14-s + 0.250·16-s + 0.0292i·17-s − 0.103i·18-s + 0.252·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(139.738\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :21/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.158460385\)
\(L(\frac12)\) \(\approx\) \(1.158460385\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.02e3iT \)
5 \( 1 \)
good3 \( 1 - 9.45e4iT - 1.04e10T^{2} \)
7 \( 1 - 3.65e8iT - 5.58e17T^{2} \)
11 \( 1 + 3.92e10T + 7.40e21T^{2} \)
13 \( 1 - 4.64e11iT - 2.47e23T^{2} \)
17 \( 1 - 2.43e11iT - 6.90e25T^{2} \)
19 \( 1 - 6.74e12T + 7.14e26T^{2} \)
23 \( 1 - 1.66e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.85e15T + 5.13e30T^{2} \)
31 \( 1 - 3.48e15T + 2.08e31T^{2} \)
37 \( 1 - 3.87e15iT - 8.55e32T^{2} \)
41 \( 1 - 1.53e17T + 7.38e33T^{2} \)
43 \( 1 - 6.75e15iT - 2.00e34T^{2} \)
47 \( 1 + 1.11e17iT - 1.30e35T^{2} \)
53 \( 1 - 2.06e18iT - 1.62e36T^{2} \)
59 \( 1 + 5.84e18T + 1.54e37T^{2} \)
61 \( 1 - 4.90e18T + 3.10e37T^{2} \)
67 \( 1 - 1.35e19iT - 2.22e38T^{2} \)
71 \( 1 - 3.28e19T + 7.52e38T^{2} \)
73 \( 1 + 3.25e19iT - 1.34e39T^{2} \)
79 \( 1 + 5.11e19T + 7.08e39T^{2} \)
83 \( 1 + 2.02e20iT - 1.99e40T^{2} \)
89 \( 1 + 2.17e20T + 8.65e40T^{2} \)
97 \( 1 + 1.03e21iT - 5.27e41T^{2} \)
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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

−11.66426945964152643228699949815, −10.73567498090375262565688541002, −9.656413085724971645528131360125, −8.996737245042436773169433356651, −7.47929051549521051325634424864, −5.76405864462371495236240344508, −4.64248494486729775283893315171, −3.74607115944084311093802291181, −2.52576236325802506291892058353, −1.33461949110581972850553900841, 0.25557325170590954961116539963, 1.09694570829464008847895853171, 2.52670115372152460500114173734, 4.03517432018293647911478264056, 5.36724287778814482388957857152, 6.51233818872039417175017756994, 7.48962168875282752912847986961, 8.168672467214365455370561458441, 9.688180936362496841401902083272, 10.85622731073376061694582720516

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