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Properties

Label 2-4650-5.4-c1-0-89
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s − 4-s + 6-s − 2i·7-s + i·8-s − 9-s − i·12-s − 4i·13-s − 2·14-s + 16-s − 6i·17-s + i·18-s − 8·19-s + 2·21-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 0.288i·12-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 1.83·19-s + 0.436·21-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: \(1\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.062670105602441463597717286282, −7.14321032128348522432913342231, −6.32248339035850254343883425160, −5.36588252718478460389381977373, −4.65105896363390711339758951542, −4.05564065479600615001790893534, −3.12284525023169260609529683631, −2.47778232823586717624063479537, −1.09164105958290793155261754246, 0, 1.71203187139770521161832689279, 2.32127349526700651926421536744, 3.72128209297662734686617066583, 4.34443499827142356908445167701, 5.37708080647256413177393891620, 6.06318987527279162976956504221, 6.59214486106846861547912984540, 7.21013462716322092885496198012, 8.165283760791482819760944968820

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