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Properties

Label 2-1200-15.2-c1-0-9
Degree $2$
Conductor $1200$
Sign $-0.655 - 0.755i$
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  + (0.448 + 1.67i)3-s + (2.44 + 2.44i)7-s + (−2.59 + 1.50i)9-s + 5.19i·11-s + (2.12 − 2.12i)17-s + i·19-s + (−3 + 5.19i)21-s + (−4.24 − 4.24i)23-s + (−3.67 − 3.67i)27-s + 2·31-s + (−8.69 + 2.32i)33-s + (2.44 + 2.44i)37-s + 5.19i·41-s + (−2.44 + 2.44i)43-s + 4.99i·49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.925 + 0.925i)7-s + (−0.866 + 0.5i)9-s + 1.56i·11-s + (0.514 − 0.514i)17-s + 0.229i·19-s + (−0.654 + 1.13i)21-s + (−0.884 − 0.884i)23-s + (−0.707 − 0.707i)27-s + 0.359·31-s + (−1.51 + 0.405i)33-s + (0.402 + 0.402i)37-s + 0.811i·41-s + (−0.373 + 0.373i)43-s + 0.714i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.655 - 0.755i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738041251\)
\(L(\frac12)\) \(\approx\) \(1.738041251\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.448 - 1.67i)T \)
5 \( 1 \)
good7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-2.12 + 2.12i)T - 17iT^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (2.44 - 2.44i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.12 + 6.12i)T - 73iT^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + (-2.12 - 2.12i)T + 83iT^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−9.820086869251708543514295491561, −9.476142547918347768923336218148, −8.311774548440590873684726946810, −7.970360827722158906957854107962, −6.70437269145692595820673320529, −5.54557253308162306735130891665, −4.84216085224071073330664287272, −4.18552700958132938622880060015, −2.80317407949117363059235309587, −1.91130208174629345261707683904, 0.73228558752995074721702968203, 1.77815878357079756629660900659, 3.16957649012807775765798560959, 4.03717477391817073978773829894, 5.41876423266923229604982971199, 6.11819096299443982342307021467, 7.11792414520452496615711797604, 7.945666516380273909717689759734, 8.296959516232969733719051913524, 9.273531273514638122646992674980

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