Properties

Label 2-50-5.4-c21-0-23
Degree $2$
Conductor $50$
Sign $0.447 + 0.894i$
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 + 6.26e4i·3-s − 1.04e6·4-s + 6.41e7·6-s + 1.35e9i·7-s + 1.07e9i·8-s + 6.53e9·9-s + 6.02e10·11-s − 6.56e10i·12-s − 6.48e11i·13-s + 1.39e12·14-s + 1.09e12·16-s − 1.36e12i·17-s − 6.69e12i·18-s − 3.73e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.612i·3-s − 0.5·4-s + 0.432·6-s + 1.81i·7-s + 0.353i·8-s + 0.625·9-s + 0.699·11-s − 0.306i·12-s − 1.30i·13-s + 1.28·14-s + 0.250·16-s − 0.164i·17-s − 0.442i·18-s − 1.39·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
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: \((2,\ 50,\ (\ :21/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.802566118\)
\(L(\frac12)\) \(\approx\) \(1.802566118\)
\(L(\frac{23}{2})\) 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 \( 1 - 6.26e4iT - 1.04e10T^{2} \)
7 \( 1 - 1.35e9iT - 5.58e17T^{2} \)
11 \( 1 - 6.02e10T + 7.40e21T^{2} \)
13 \( 1 + 6.48e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.36e12iT - 6.90e25T^{2} \)
19 \( 1 + 3.73e13T + 7.14e26T^{2} \)
23 \( 1 + 3.31e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.94e15T + 5.13e30T^{2} \)
31 \( 1 - 6.11e15T + 2.08e31T^{2} \)
37 \( 1 + 2.13e16iT - 8.55e32T^{2} \)
41 \( 1 - 7.53e16T + 7.38e33T^{2} \)
43 \( 1 - 1.11e17iT - 2.00e34T^{2} \)
47 \( 1 + 3.03e17iT - 1.30e35T^{2} \)
53 \( 1 + 1.33e18iT - 1.62e36T^{2} \)
59 \( 1 - 8.63e17T + 1.54e37T^{2} \)
61 \( 1 - 4.07e18T + 3.10e37T^{2} \)
67 \( 1 + 2.05e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.84e19T + 7.52e38T^{2} \)
73 \( 1 - 8.35e18iT - 1.34e39T^{2} \)
79 \( 1 - 2.13e19T + 7.08e39T^{2} \)
83 \( 1 - 1.83e20iT - 1.99e40T^{2} \)
89 \( 1 - 3.15e20T + 8.65e40T^{2} \)
97 \( 1 - 1.08e20iT - 5.27e41T^{2} \)
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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

−11.13972610958700052012564713155, −10.09910635493778632368484932358, −9.103193027792982625738073265970, −8.269162924029932205775474391769, −6.31522854843994581924976369014, −5.19143480404494016748543031095, −4.11215448759165473003066603105, −2.82836249767792263038667234949, −1.94396462984676041317479021661, −0.42478016174847284541773367981, 0.924468923084475613174474882522, 1.70213287154666229541442789034, 3.89999331597547030439088534111, 4.39355531713772162724017400517, 6.30975675139631440483528511811, 7.03560417721999612065268606715, 7.75762371502049936101248157214, 9.248520553893118535867769865738, 10.34681839947533851418839983711, 11.64114949084399878130466295771

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