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Properties

Label 2-1200-15.2-c1-0-6
Degree $2$
Conductor $1200$
Sign $-0.749 - 0.662i$
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 + (3 + 3i)7-s + (−2.82 − i)9-s − 1.41i·11-s + (−4.24 + 4.24i)17-s + 4i·19-s + (−5.99 + 4.24i)21-s + (2.82 + 2.82i)23-s + (2.53 − 4.53i)27-s + 1.41·29-s + 2·31-s + (2.41 + 0.414i)33-s + (2 + 2i)37-s − 5.65i·41-s + (−2 + 2i)43-s + ⋯
L(s)  = 1  + (−0.169 + 0.985i)3-s + (1.13 + 1.13i)7-s + (−0.942 − 0.333i)9-s − 0.426i·11-s + (−1.02 + 1.02i)17-s + 0.917i·19-s + (−1.30 + 0.925i)21-s + (0.589 + 0.589i)23-s + (0.487 − 0.872i)27-s + 0.262·29-s + 0.359·31-s + (0.420 + 0.0721i)33-s + (0.328 + 0.328i)37-s − 0.883i·41-s + (−0.304 + 0.304i)43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.749 - 0.662i$
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: \((2,\ 1200,\ (\ :1/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.395006616\)
\(L(\frac12)\) \(\approx\) \(1.395006616\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-2 - 2i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + (2.82 + 2.82i)T + 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-13 - 13i)T + 97iT^{2} \)
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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

−10.05902522573125798598781703481, −9.181914116916814835516695050350, −8.479618069707434351060621477267, −8.015520197236927409889957202088, −6.45808988721811816930781415782, −5.69671394688587711301125808722, −4.97159001585485841619855820413, −4.11467524151267280266728841009, −2.98479601770405182424978645371, −1.75324699816212045526266982568, 0.61355404920622851044851102891, 1.79541117466357769325734428489, 2.91608857919995405952627524945, 4.55563619792721296359389216392, 4.92568993109496884404957976731, 6.36195223009988681703948942208, 7.06829887624432967768639841045, 7.61780310662649722462491500672, 8.449625273560461429269348567815, 9.292240294339336693782406359400

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