Properties

Label 2-4650-5.4-c1-0-19
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 − 4i·7-s i·8-s − 9-s − 4·11-s i·12-s + 2i·13-s + 4·14-s + 16-s − 2i·17-s i·18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s + 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s − 0.917·19-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

\(L(1)\) \(\approx\) \(1.080969779\)
\(L(\frac12)\) \(\approx\) \(1.080969779\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−8.394022292481085227518729019548, −7.75475318147133919743262546715, −7.12948124301263205297912561495, −6.51048945565580877858949820364, −5.51165096256350137043810090187, −4.85697016481569259983874591109, −4.15747629675926135435545382868, −3.52132930027113168396251459676, −2.34288383281389928116130857484, −0.812648249560500697296726980955, 0.38996351335477283426337970550, 1.86814498157576651718195197531, 2.53050988853198620136101480404, 3.06142356309532815642618560495, 4.35348170402238053517335743892, 5.15851472978656081536133065093, 5.89798258113482087571171974761, 6.36120002142041072559448162399, 7.73997882823514993689838467110, 8.022034066296966564146756463756

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