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Properties

Label 2-50-5.4-c19-0-19
Degree $2$
Conductor $50$
Sign $-0.894 + 0.447i$
Analytic cond. $114.408$
Root an. cond. $10.6961$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 512i·2-s − 1.73e4i·3-s − 2.62e5·4-s − 8.88e6·6-s − 5.57e7i·7-s + 1.34e8i·8-s + 8.61e8·9-s − 5.35e9·11-s + 4.54e9i·12-s + 6.91e10i·13-s − 2.85e10·14-s + 6.87e10·16-s − 8.22e11i·17-s − 4.40e11i·18-s + 3.19e11·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.508i·3-s − 0.5·4-s − 0.359·6-s − 0.522i·7-s + 0.353i·8-s + 0.741·9-s − 0.685·11-s + 0.254i·12-s + 1.80i·13-s − 0.369·14-s + 0.250·16-s − 1.68i·17-s − 0.524i·18-s + 0.227·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(114.408\)
Root analytic conductor: \(10.6961\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :19/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.730862605\)
\(L(\frac12)\) \(\approx\) \(1.730862605\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 512iT \)
5 \( 1 \)
good3 \( 1 + 1.73e4iT - 1.16e9T^{2} \)
7 \( 1 + 5.57e7iT - 1.13e16T^{2} \)
11 \( 1 + 5.35e9T + 6.11e19T^{2} \)
13 \( 1 - 6.91e10iT - 1.46e21T^{2} \)
17 \( 1 + 8.22e11iT - 2.39e23T^{2} \)
19 \( 1 - 3.19e11T + 1.97e24T^{2} \)
23 \( 1 - 8.01e12iT - 7.46e25T^{2} \)
29 \( 1 + 2.41e13T + 6.10e27T^{2} \)
31 \( 1 - 2.07e14T + 2.16e28T^{2} \)
37 \( 1 - 2.90e14iT - 6.24e29T^{2} \)
41 \( 1 - 9.57e14T + 4.39e30T^{2} \)
43 \( 1 + 5.00e15iT - 1.08e31T^{2} \)
47 \( 1 - 3.13e14iT - 5.88e31T^{2} \)
53 \( 1 + 1.16e16iT - 5.77e32T^{2} \)
59 \( 1 - 2.11e16T + 4.42e33T^{2} \)
61 \( 1 - 4.20e16T + 8.34e33T^{2} \)
67 \( 1 + 2.45e17iT - 4.95e34T^{2} \)
71 \( 1 + 3.98e17T + 1.49e35T^{2} \)
73 \( 1 + 5.53e17iT - 2.53e35T^{2} \)
79 \( 1 - 1.76e18T + 1.13e36T^{2} \)
83 \( 1 - 1.89e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.56e18T + 1.09e37T^{2} \)
97 \( 1 + 1.22e19iT - 5.60e37T^{2} \)
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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

−11.40603885863003381459702730950, −10.07944784570353570333446431724, −9.189879040668043274111929115413, −7.61833166595229432950179686904, −6.77086773895658191399135554089, −4.99645635141766035807604281441, −3.97684252973236866686426127814, −2.51719797748099171836364028877, −1.48044444954281655570949487487, −0.43118998061997429984035201457, 0.996540123565983341334007289321, 2.73242016353571775501212404654, 4.07133480292001590559103890102, 5.23836378639303074355942443853, 6.18375373971857637843275841181, 7.71151566435928984158117542335, 8.522880989973663034171146681333, 9.982121620271125376420680747901, 10.64413176245405877011843200851, 12.54737843735406583259789300338

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