Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.856 + 0.516i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 0.264i)2-s − 3-s + (1.85 − 0.735i)4-s + (−1.38 + 0.264i)6-s − 0.941i·7-s + (2.38 − 1.51i)8-s + 9-s + 4.49i·11-s + (−1.85 + 0.735i)12-s + 5.55·13-s + (−0.249 − 1.30i)14-s + (2.91 − 2.73i)16-s − 7.55i·17-s + (1.38 − 0.264i)18-s − 1.05i·19-s + ⋯
L(s)  = 1  + (0.982 − 0.187i)2-s − 0.577·3-s + (0.929 − 0.367i)4-s + (−0.567 + 0.108i)6-s − 0.355i·7-s + (0.844 − 0.535i)8-s + 0.333·9-s + 1.35i·11-s + (−0.536 + 0.212i)12-s + 1.54·13-s + (−0.0665 − 0.349i)14-s + (0.729 − 0.683i)16-s − 1.83i·17-s + (0.327 − 0.0623i)18-s − 0.242i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.856 + 0.516i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ 0.856 + 0.516i)\)
\(L(1)\)  \(\approx\)  \(2.33801 - 0.650078i\)
\(L(\frac12)\)  \(\approx\)  \(2.33801 - 0.650078i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.38 + 0.264i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 0.941iT - 7T^{2} \)
11 \( 1 - 4.49iT - 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 + 7.55iT - 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 + 1.05iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 - 1.88T + 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 8.49iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 5.88T + 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−10.72992306461491445961344029193, −10.08989519476829719831793076320, −8.989904735524230319021786178609, −7.44312380831015834111243783975, −6.90908076293703441519075710540, −5.91361269277593744391465211957, −4.89163429170202900306328328012, −4.18420043824646641370016337398, −2.86838385467434311690789796059, −1.33378189332377377405769252178, 1.59591393022744840411065286057, 3.33983476215168578230037569483, 4.05350888111005769883656762104, 5.50911098886993741796080005905, 5.99232565697478378944750800785, 6.70836859372949431436838469762, 8.180836523957290741283502798426, 8.616549389813603675800261820499, 10.34527958349583002693896307601, 10.91801270488003178423596395828

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