Properties

Label 2-1200-15.8-c1-0-23
Degree $2$
Conductor $1200$
Sign $0.991 - 0.130i$
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  + (1.22 + 1.22i)3-s + (3.67 − 3.67i)7-s + 2.99i·9-s + (1.22 + 1.22i)13-s − 7i·19-s + 9·21-s + (−3.67 + 3.67i)27-s + 11·31-s + (−4.89 + 4.89i)37-s + 2.99i·39-s + (−1.22 − 1.22i)43-s − 20i·49-s + (8.57 − 8.57i)57-s − 61-s + (11.0 + 11.0i)63-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (1.38 − 1.38i)7-s + 0.999i·9-s + (0.339 + 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (−0.707 + 0.707i)27-s + 1.97·31-s + (−0.805 + 0.805i)37-s + 0.480i·39-s + (−0.186 − 0.186i)43-s − 2.85i·49-s + (1.13 − 1.13i)57-s − 0.128·61-s + (1.38 + 1.38i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.991 - 0.130i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488754915\)
\(L(\frac12)\) \(\approx\) \(2.488754915\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-3.67 + 3.67i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1.22 - 1.22i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (8.57 - 8.57i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-9.79 - 9.79i)T + 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (3.67 - 3.67i)T - 97iT^{2} \)
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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

−9.883419415705688192408875294127, −8.797409752305538611306568622063, −8.256706726417695643030417323288, −7.44903624111486981578733417630, −6.66939362729089187202259762112, −5.05845320738339636101057930467, −4.57912296300235672015326611217, −3.77890601369810442353567788891, −2.54625793442581065754182216381, −1.21062990634165342564177201413, 1.42920356221139762698288822257, 2.25256315194500926898491817944, 3.32019184603891760414973137728, 4.59098462127085621477486890515, 5.66163786826726662510357345434, 6.29249271334468575033423357659, 7.58655141731098056298687027709, 8.200007308409463622674035543146, 8.617791554353112322188278304130, 9.506151683585505346193621786490

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