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Properties

Label 2-1200-15.8-c1-0-12
Degree $2$
Conductor $1200$
Sign $0.973 + 0.229i$
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 + (2.44 − 2.44i)7-s + 2.99i·9-s + (4.89 + 4.89i)13-s + 8i·19-s − 5.99·21-s + (3.67 − 3.67i)27-s − 4·31-s + (4.89 − 4.89i)37-s − 11.9i·39-s + (7.34 + 7.34i)43-s − 4.99i·49-s + (9.79 − 9.79i)57-s + 14·61-s + (7.34 + 7.34i)63-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.925 − 0.925i)7-s + 0.999i·9-s + (1.35 + 1.35i)13-s + 1.83i·19-s − 1.30·21-s + (0.707 − 0.707i)27-s − 0.718·31-s + (0.805 − 0.805i)37-s − 1.92i·39-s + (1.12 + 1.12i)43-s − 0.714i·49-s + (1.29 − 1.29i)57-s + 1.79·61-s + (0.925 + 0.925i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492924303\)
\(L(\frac12)\) \(\approx\) \(1.492924303\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.89 - 4.89i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-7.34 - 7.34i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (-2.44 + 2.44i)T - 67iT^{2} \)
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 \( 1 + (-9.79 + 9.79i)T - 97iT^{2} \)
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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

−9.862919084795743287044580317453, −8.707942425074359205205208186687, −7.907379914827237304591740676944, −7.30012201275111563307110199381, −6.33360383097670384700274976645, −5.69729957882976061251911394362, −4.48676846074074524970299303122, −3.81545736973331554632601859933, −1.91299801198480199484279969546, −1.16354033796269817599740498782, 0.894976030896496959444208243933, 2.58936030361621522850748381655, 3.72538329168478721571651129590, 4.83530600642989268922747141246, 5.47592838312825359066453933285, 6.14637066227117834742440675677, 7.26804563348882440188022671207, 8.469349080978693308946044805363, 8.837152012640085170406904227496, 9.835223324398076957351019596665

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