Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.563 + 0.825i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.450 + 1.34i)2-s + i·3-s + (−1.59 + 1.20i)4-s + (−1.34 + 0.450i)6-s − 2.64·7-s + (−2.33 − 1.59i)8-s − 9-s + 1.51i·11-s + (−1.20 − 1.59i)12-s + 3.87i·13-s + (−1.18 − 3.54i)14-s + (1.08 − 3.84i)16-s − 3.31·17-s + (−0.450 − 1.34i)18-s − 7.08i·19-s + ⋯
L(s)  = 1  + (0.318 + 0.947i)2-s + 0.577i·3-s + (−0.797 + 0.603i)4-s + (−0.547 + 0.183i)6-s − 0.998·7-s + (−0.825 − 0.563i)8-s − 0.333·9-s + 0.456i·11-s + (−0.348 − 0.460i)12-s + 1.07i·13-s + (−0.317 − 0.946i)14-s + (0.271 − 0.962i)16-s − 0.803·17-s + (−0.106 − 0.315i)18-s − 1.62i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.563 + 0.825i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (301, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.563 + 0.825i)\)
\(L(1)\)  \(\approx\)  \(0.277994 - 0.526388i\)
\(L(\frac12)\)  \(\approx\)  \(0.277994 - 0.526388i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.450 - 1.34i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 - 3.87iT - 13T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 + 7.08iT - 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 2.18iT - 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 - 7.87iT - 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 - 1.01iT - 43T^{2} \)
47 \( 1 - 7.08T + 47T^{2} \)
53 \( 1 - 4.50iT - 53T^{2} \)
59 \( 1 - 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 - 1.01iT - 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−11.26444153145257290644283589539, −10.10072985281646554590310143752, −9.172777691871051603627047364100, −8.900995494827857685119987846099, −7.42594845241027896663923377771, −6.72665972837522423557202747024, −5.89934040240097214639981962155, −4.67938071288689578034888146540, −4.01800411772319870696005235266, −2.72535109197844120118420219990, 0.28414314521316499865016354011, 1.99516143229671399088654309103, 3.19194387930157377635838153012, 4.04727958521448031845236741635, 5.70472500153949956481998182233, 6.05134737767511622815740880373, 7.50303560452509255348084025326, 8.478622406006694835147743953792, 9.407814095688194114873536469407, 10.25222357077603019375470291116

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