Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.831 - 0.555i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 0.0591i)2-s − 3-s + (1.99 − 0.167i)4-s + (1.41 − 0.0591i)6-s + 1.33i·7-s + (−2.80 + 0.353i)8-s + 9-s − 2.94i·11-s + (−1.99 + 0.167i)12-s + 2.04·13-s + (−0.0788 − 1.88i)14-s + (3.94 − 0.665i)16-s + 3.61i·17-s + (−1.41 + 0.0591i)18-s + 5.35i·19-s + ⋯
L(s)  = 1  + (−0.999 + 0.0418i)2-s − 0.577·3-s + (0.996 − 0.0835i)4-s + (0.576 − 0.0241i)6-s + 0.504i·7-s + (−0.992 + 0.125i)8-s + 0.333·9-s − 0.887i·11-s + (−0.575 + 0.0482i)12-s + 0.566·13-s + (−0.0210 − 0.503i)14-s + (0.986 − 0.166i)16-s + 0.876i·17-s + (−0.333 + 0.0139i)18-s + 1.22i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.831 - 0.555i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ 0.831 - 0.555i)\)
\(L(1)\)  \(\approx\)  \(0.731645 + 0.221956i\)
\(L(\frac12)\)  \(\approx\)  \(0.731645 + 0.221956i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.41 - 0.0591i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 - 1.33iT - 7T^{2} \)
11 \( 1 + 2.94iT - 11T^{2} \)
13 \( 1 - 2.04T + 13T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 - 5.35iT - 19T^{2} \)
23 \( 1 + 8.59iT - 23T^{2} \)
29 \( 1 - 5.26iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 - 6.55T + 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 - 9.97iT - 47T^{2} \)
53 \( 1 - 6.12T + 53T^{2} \)
59 \( 1 - 4.75iT - 59T^{2} \)
61 \( 1 - 8.51iT - 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 2.62T + 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−10.78417648825096905461549113789, −9.973471925001616319216447697484, −8.821539298158027074927332579530, −8.358744065000573032624858142218, −7.27685471614451363071515486156, −6.01357109097480799142440482788, −5.87123892464566269895121110986, −4.05528042263026462920474087232, −2.62346636959426063065352770316, −1.10027247022664126195243664359, 0.799521571847183911392092541634, 2.31734154628639968216076160824, 3.85735475014325986404458960568, 5.18807823905915805433305791122, 6.28899622935294998203927862791, 7.25549654138229578244611706389, 7.70529044987867028768695249597, 9.145565358490800521737378010825, 9.611884022627354372403543022072, 10.57350908731239145153630599142

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