Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.988 + 0.151i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.576 − 1.29i)2-s − 3-s + (−1.33 + 1.48i)4-s + (0.576 + 1.29i)6-s − 1.97i·7-s + (2.69 + 0.867i)8-s + 9-s + 1.43i·11-s + (1.33 − 1.48i)12-s − 0.241·13-s + (−2.55 + 1.13i)14-s + (−0.430 − 3.97i)16-s − 7.38i·17-s + (−0.576 − 1.29i)18-s + 3.04i·19-s + ⋯
L(s)  = 1  + (−0.407 − 0.913i)2-s − 0.577·3-s + (−0.667 + 0.744i)4-s + (0.235 + 0.527i)6-s − 0.747i·7-s + (0.951 + 0.306i)8-s + 0.333·9-s + 0.431i·11-s + (0.385 − 0.429i)12-s − 0.0669·13-s + (−0.682 + 0.304i)14-s + (−0.107 − 0.994i)16-s − 1.79i·17-s + (−0.135 − 0.304i)18-s + 0.697i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.988 + 0.151i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.988 + 0.151i)\)
\(L(1)\)  \(\approx\)  \(0.0416357 - 0.547384i\)
\(L(\frac12)\)  \(\approx\)  \(0.0416357 - 0.547384i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.576 + 1.29i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 + 0.241T + 13T^{2} \)
17 \( 1 + 7.38iT - 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 + 0.874iT - 23T^{2} \)
29 \( 1 + 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + 8.81T + 37T^{2} \)
41 \( 1 + 1.91T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 3.34iT - 47T^{2} \)
53 \( 1 - 9.20T + 53T^{2} \)
59 \( 1 + 6.43iT - 59T^{2} \)
61 \( 1 + 4.57iT - 61T^{2} \)
67 \( 1 + 4.86T + 67T^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 - 4.12iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−10.15904752119304421374262218608, −9.783016757802585121928541165533, −8.678557615193557309162527332370, −7.54806652950994005058301993775, −6.94437067878494030250561097370, −5.38164773133632364656506531603, −4.46645372898650948216136211713, −3.43344876378660497944658171183, −1.95201572928151789800875427303, −0.38538771242626630338935628688, 1.63180324251250248226735114995, 3.66650105086487649643329325288, 5.03467278603935347831972202236, 5.70127714066814780349967657830, 6.55462898895237448984203107038, 7.42088177625559658341959099886, 8.648408908981043639484675445913, 8.949937040414259038824935878229, 10.29891275578917837694449881964, 10.76406171127108198851658645260

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