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Properties

Label 2-1200-15.14-c2-0-3
Degree $2$
Conductor $1200$
Sign $-0.447 + 0.894i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 11i·7-s − 9·9-s − i·13-s − 37·19-s − 33·21-s − 27i·27-s + 13·31-s − 26i·37-s + 3·39-s + 61i·43-s − 72·49-s − 111i·57-s + 47·61-s − 99i·63-s + ⋯
L(s)  = 1  + i·3-s + 1.57i·7-s − 9-s − 0.0769i·13-s − 1.94·19-s − 1.57·21-s − i·27-s + 0.419·31-s − 0.702i·37-s + 0.0769·39-s + 1.41i·43-s − 1.46·49-s − 1.94i·57-s + 0.770·61-s − 1.57i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4064754812\)
\(L(\frac12)\) \(\approx\) \(0.4064754812\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 - 11iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 37T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 13T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 61iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 47T + 3.72e3T^{2} \)
67 \( 1 + 109iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46iT - 5.32e3T^{2} \)
79 \( 1 + 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 169iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03279236416506976452178635254, −9.212100327782731165215996673394, −8.666204575580779904601737276446, −8.006294765497300309355320515595, −6.46615418144426390652904901100, −5.86487680882317562721923552610, −4.98472181945554782699013080949, −4.14654345515202671230844365506, −2.95597746423491641131145897596, −2.12316299742157252654822115106, 0.12081445406674269551004232368, 1.24790171357949941773531232964, 2.41018242442888201137272884183, 3.72017325865228470703486745269, 4.56234939643285086739954934317, 5.83943329495695939392231193044, 6.77517813698799847006260247496, 7.16064791986092684906923435188, 8.159715674917965679502245349895, 8.709317208802279801979307369268

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