Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.971 + 0.237i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.796 + 1.16i)2-s − 3-s + (−0.731 + 1.86i)4-s + (−0.796 − 1.16i)6-s − 4.72i·7-s + (−2.75 + 0.627i)8-s + 9-s − 3.93i·11-s + (0.731 − 1.86i)12-s + 3.46·13-s + (5.51 − 3.76i)14-s + (−2.93 − 2.72i)16-s − 3.51i·17-s + (0.796 + 1.16i)18-s + 5.44i·19-s + ⋯
L(s)  = 1  + (0.563 + 0.826i)2-s − 0.577·3-s + (−0.365 + 0.930i)4-s + (−0.325 − 0.477i)6-s − 1.78i·7-s + (−0.975 + 0.221i)8-s + 0.333·9-s − 1.18i·11-s + (0.211 − 0.537i)12-s + 0.961·13-s + (1.47 − 1.00i)14-s + (−0.732 − 0.680i)16-s − 0.852i·17-s + (0.187 + 0.275i)18-s + 1.24i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.971 + 0.237i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ 0.971 + 0.237i)\)
\(L(1)\)  \(\approx\)  \(1.40395 - 0.169161i\)
\(L(\frac12)\)  \(\approx\)  \(1.40395 - 0.169161i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.796 - 1.16i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 4.72iT - 7T^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 3.51iT - 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + 0.414T + 37T^{2} \)
41 \( 1 - 3.00T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 - 0.925iT - 47T^{2} \)
53 \( 1 - 0.233T + 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 + 0.118iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 0.563iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 8.88T + 89T^{2} \)
97 \( 1 + 7.27iT - 97T^{2} \)
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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

−10.78940821858087796767321753899, −9.871738201482319245387567324551, −8.504991852577204139014190162819, −7.86412382185461724500452979005, −6.76950977940800217771715952274, −6.28043033993528495516403266436, −5.12232335622226945838883861374, −4.11684940789948159194716655586, −3.34857366994614802628597461169, −0.75378782381264925568275376412, 1.65741592240601726137252813628, 2.73613347639555669539913242745, 4.12944620590399718608812948411, 5.18888928974745412117083229895, 5.85020378249386504780783326119, 6.73463306743801881842293650224, 8.322106292135072679998346309584, 9.247816033024984076394132220388, 9.874909655210862644928444077086, 10.98562098588897618690806347033

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