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Properties

Label 2-1200-15.2-c1-0-1
Degree $2$
Conductor $1200$
Sign $-0.998 + 0.0618i$
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 + 1.41i)3-s + (−2.41 − 2.41i)7-s + (−1.00 + 2.82i)9-s − 0.828i·11-s + (−3.82 + 3.82i)13-s + (−1.82 + 1.82i)17-s + 0.828i·19-s + (1 − 5.82i)21-s + (−4.41 − 4.41i)23-s + (−5.00 + 1.41i)27-s − 3.65·29-s − 5.65·31-s + (1.17 − 0.828i)33-s + (5.82 + 5.82i)37-s + (−9.24 − 1.58i)39-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (−0.912 − 0.912i)7-s + (−0.333 + 0.942i)9-s − 0.249i·11-s + (−1.06 + 1.06i)13-s + (−0.443 + 0.443i)17-s + 0.190i·19-s + (0.218 − 1.27i)21-s + (−0.920 − 0.920i)23-s + (−0.962 + 0.272i)27-s − 0.679·29-s − 1.01·31-s + (0.203 − 0.144i)33-s + (0.958 + 0.958i)37-s + (−1.48 − 0.253i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
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: \((2,\ 1200,\ (\ :1/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4528669564\)
\(L(\frac12)\) \(\approx\) \(0.4528669564\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (2.41 + 2.41i)T + 7iT^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (3.82 - 3.82i)T - 13iT^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (4.41 + 4.41i)T + 23iT^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-5.82 - 5.82i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (0.414 - 0.414i)T - 43iT^{2} \)
47 \( 1 + (3.58 - 3.58i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (-10.0 - 10.0i)T + 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-4.65 + 4.65i)T - 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (1 + i)T + 97iT^{2} \)
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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

−9.811437404485013409791317742860, −9.684446016282307923861847762488, −8.638115406578335728289860886696, −7.77802864289136272662669490283, −6.89920405406967857558751139335, −6.05220381751170670190642923638, −4.71666115217938112836416365514, −4.10495002767833472807711009460, −3.20564327021690093252040719150, −2.05413978950888941734754582229, 0.16171795013699758381330114852, 2.06332024960361941121247990895, 2.80775495534921349368565919216, 3.78024266868612202801429035816, 5.33568015802922598492884484575, 5.97054122278278134411115808651, 7.02558053219048857656282060789, 7.60396984266122116427741441470, 8.487824522643052921925487546365, 9.525922842401193926037709174176

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