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Properties

Label 2-1200-15.8-c1-0-25
Degree $2$
Conductor $1200$
Sign $0.662 + 0.749i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.292 + 1.70i)3-s + (2 − 2i)7-s + (−2.82 + i)9-s − 5.65i·11-s + (−2.82 − 2.82i)17-s − 4i·19-s + (4 + 2.82i)21-s + (4.24 − 4.24i)23-s + (−2.53 − 4.53i)27-s − 5.65·29-s − 8·31-s + (9.65 − 1.65i)33-s + (8 − 8i)37-s + 5.65i·41-s + (2 + 2i)43-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (0.755 − 0.755i)7-s + (−0.942 + 0.333i)9-s − 1.70i·11-s + (−0.685 − 0.685i)17-s − 0.917i·19-s + (0.872 + 0.617i)21-s + (0.884 − 0.884i)23-s + (−0.487 − 0.872i)27-s − 1.05·29-s − 1.43·31-s + (1.68 − 0.288i)33-s + (1.31 − 1.31i)37-s + 0.883i·41-s + (0.304 + 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.501489683\)
\(L(\frac12)\) \(\approx\) \(1.501489683\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 \)
good7 \( 1 + (-2 + 2i)T - 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-8 + 8i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + (5.65 - 5.65i)T - 53iT^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-6 + 6i)T - 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + (-8 - 8i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (-9.89 + 9.89i)T - 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (8 - 8i)T - 97iT^{2} \)
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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

−9.425000735199989988386988763348, −8.989461607300445602265481316977, −8.147052299111742107454138284840, −7.30176268630976462455590035633, −6.14045354495691804323772519623, −5.21088399746162834604688941721, −4.44310118815521598289723043565, −3.54115057876037472306157906423, −2.53579981867651306039958508650, −0.62352581789061782803589833771, 1.68278238994317557239708880295, 2.15120990843273350105846486230, 3.62329009957845013216448332342, 4.87869756363237281080463613951, 5.67939968546028666580925107230, 6.66371169786463120612432130959, 7.49353773451761846038010546574, 8.045029844321695432031469039923, 9.010987494208007751155033870945, 9.616605346630173676164080367940

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