Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.247 - 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.806 − 1.16i)2-s + i·3-s + (−0.699 + 1.87i)4-s + (1.16 − 0.806i)6-s − 0.746·7-s + (2.74 − 0.699i)8-s − 9-s + 5.36i·11-s + (−1.87 − 0.699i)12-s − 2.92i·13-s + (0.601 + 0.866i)14-s + (−3.02 − 2.62i)16-s − 2.13·17-s + (0.806 + 1.16i)18-s + 1.73i·19-s + ⋯
L(s)  = 1  + (−0.570 − 0.821i)2-s + 0.577i·3-s + (−0.349 + 0.936i)4-s + (0.474 − 0.329i)6-s − 0.282·7-s + (0.968 − 0.247i)8-s − 0.333·9-s + 1.61i·11-s + (−0.540 − 0.201i)12-s − 0.811i·13-s + (0.160 + 0.231i)14-s + (−0.755 − 0.655i)16-s − 0.517·17-s + (0.190 + 0.273i)18-s + 0.397i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.247 - 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (301, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.247 - 0.968i)\)
\(L(1)\)  \(\approx\)  \(0.342859 + 0.441297i\)
\(L(\frac12)\)  \(\approx\)  \(0.342859 + 0.441297i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.806 + 1.16i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 0.746T + 7T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + 2.92iT - 13T^{2} \)
17 \( 1 + 2.13T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 7.49T + 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 - 1.07iT - 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 - 7.72iT - 53T^{2} \)
59 \( 1 - 6.85iT - 59T^{2} \)
61 \( 1 + 6.45iT - 61T^{2} \)
67 \( 1 - 7.44iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 0.690T + 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 + 5.85iT - 83T^{2} \)
89 \( 1 - 7.59T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.62675341297917817811957433306, −10.04863750691141834805443638203, −9.514631783576072444313604290402, −8.448739400445218208854029071417, −7.66007679753556324749607342123, −6.56886070397162498410462174341, −5.05602147133902731873060567504, −4.18085000834518565530146837729, −3.09348413190969811732507335812, −1.83098214645737332572255802108, 0.36412827931720773684822848800, 2.05918041610614081917854046339, 3.77950276141966691808917156716, 5.17890563335920392732985750429, 6.27914015174090946836960195394, 6.61940670336845572818184136441, 7.915092357693926310956011629056, 8.457819602683363182750781745326, 9.316793443123483233770848648248, 10.24913880386126223497420284432

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