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  =  \((2,\ 600,\ (\ :1/2),\ -0.247 + 0.968i)\)
\(L(1)\)  \(\approx\)  \(1.09650 - 1.41132i\)
\(L(\frac12)\)  \(\approx\)  \(1.09650 - 1.41132i\)
\(L(\frac{3}{2})\)   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

−10.61707880311331032738830003985, −9.773013455571653581663699652960, −8.841572055251705857021683828748, −8.051358120419233909427017440717, −6.42806484619486163230968327003, −5.52931135472097781278342657109, −4.77413096614882401058908058219, −3.50054771203984655960273502930, −2.82148910602230004932074902259, −0.881974301390377661719713125968, 1.88015672129122475097908214960, 3.36937023508344912732079935475, 4.66382880803750687960697947190, 5.32946146228741333314779173856, 6.78299399252938054796200938983, 7.02205000635388326457640921666, 8.049172424857950598677491643532, 8.937839683200887053477301436268, 9.856593959508064893103253404790, 11.15439272160168492547802129855

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