Properties

Label 2-4650-5.4-c1-0-48
Degree $2$
Conductor $4650$
Sign $0.447 + 0.894i$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 6-s − 3.37i·7-s + i·8-s − 9-s + 0.627·11-s i·12-s − 2i·13-s − 3.37·14-s + 16-s + 4.74i·17-s + i·18-s + 0.627·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.27i·7-s + 0.353i·8-s − 0.333·9-s + 0.189·11-s − 0.288i·12-s − 0.554i·13-s − 0.901·14-s + 0.250·16-s + 1.15i·17-s + 0.235i·18-s + 0.144·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 31\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(37.1304\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4650} (3349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.720502005\)
\(L(\frac12)\) \(\approx\) \(1.720502005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 \)
31 \( 1 + T \)
good7 \( 1 + 3.37iT - 7T^{2} \)
11 \( 1 - 0.627T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 4.74iT - 17T^{2} \)
19 \( 1 - 0.627T + 19T^{2} \)
23 \( 1 - 3.37iT - 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
37 \( 1 - 0.744iT - 37T^{2} \)
41 \( 1 - 0.744T + 41T^{2} \)
43 \( 1 - 0.627iT - 43T^{2} \)
47 \( 1 - 6.74iT - 47T^{2} \)
53 \( 1 - 1.37iT - 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 3.37T + 71T^{2} \)
73 \( 1 + 8.11iT - 73T^{2} \)
79 \( 1 - 4.62T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 1.37T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.192017536045493867950268112862, −7.68984420582537060259168678807, −6.69394176156565045243946232568, −5.91421789864177213277709648311, −4.95636753391204270425545821957, −4.28061554501289653605996187645, −3.63321904503824463639674719434, −2.94818821688529834835537399911, −1.65074247318675913164889480236, −0.66313467354204120915462330711, 0.836845891708379972901534893699, 2.20687777356438528072049199590, 2.86188752524180018574005077648, 4.06590189310268286276452468011, 5.06264363623654079150035824049, 5.50333201654178268739340122615, 6.51575954451201944463177475635, 6.77123638935117069321926595570, 7.69042602770938014286757236363, 8.487683533917226468506617671636

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