Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.929 + 0.368i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.190 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.500 + 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − 18-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.190 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.500 + 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − 18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.929 + 0.368i$
motivic weight  =  \(0\)
character  :  $\chi_{600} (581, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 600,\ (\ :0),\ -0.929 + 0.368i)$
$L(\frac{1}{2})$  $\approx$  $0.7250359177$
$L(\frac12)$  $\approx$  $0.7250359177$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
good7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.14026000249862335144895784802, −9.693507600777819284632008893034, −8.669158990471229483665836356675, −8.337536421615635499183523052506, −7.14254561741020520523290292026, −6.01834080393432966584567233959, −4.46165948994787499370132581884, −3.43394796637693852198834991532, −2.62389902806177915964267520957, −0.890317674643063229284811383072, 2.70509528835814710740052894529, 3.67476107723092059868908245222, 4.73415586335395170227535938226, 6.16801099675516020279824416309, 6.86649025843642921970021235882, 7.66981501816775883330459380997, 8.620230155859547483875702367982, 9.580470897731809417451843315092, 10.07883650341338401472111044932, 10.69806784290767850885714840162

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