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Properties

Label 2-1200-15.2-c1-0-29
Degree $2$
Conductor $1200$
Sign $0.0618 + 0.998i$
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  + (1.41 − i)3-s + (0.414 + 0.414i)7-s + (1.00 − 2.82i)9-s − 4.82i·11-s + (1.82 − 1.82i)13-s + (−3.82 + 3.82i)17-s − 4.82i·19-s + (1 + 0.171i)21-s + (1.58 + 1.58i)23-s + (−1.41 − 5.00i)27-s − 7.65·29-s + 5.65·31-s + (−4.82 − 6.82i)33-s + (0.171 + 0.171i)37-s + (0.757 − 4.41i)39-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + (0.156 + 0.156i)7-s + (0.333 − 0.942i)9-s − 1.45i·11-s + (0.507 − 0.507i)13-s + (−0.928 + 0.928i)17-s − 1.10i·19-s + (0.218 + 0.0374i)21-s + (0.330 + 0.330i)23-s + (−0.272 − 0.962i)27-s − 1.42·29-s + 1.01·31-s + (−0.840 − 1.18i)33-s + (0.0282 + 0.0282i)37-s + (0.121 − 0.706i)39-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.099823530\)
\(L(\frac12)\) \(\approx\) \(2.099823530\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
5 \( 1 \)
good7 \( 1 + (-0.414 - 0.414i)T + 7iT^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + (-1.82 + 1.82i)T - 13iT^{2} \)
17 \( 1 + (3.82 - 3.82i)T - 17iT^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (-1.58 - 1.58i)T + 23iT^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-2.41 + 2.41i)T - 43iT^{2} \)
47 \( 1 + (-6.41 + 6.41i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (4.07 + 4.07i)T + 67iT^{2} \)
71 \( 1 + 6.48iT - 71T^{2} \)
73 \( 1 + (6.65 - 6.65i)T - 73iT^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 + (5.24 + 5.24i)T + 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
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

−9.218379066010054479299691968807, −8.642602807663021382981388086478, −8.140258155825851984299967618173, −7.11366403377253658559805787562, −6.29642219959939020306772271874, −5.48136026305845552081475456766, −4.06637819543971443540780013244, −3.23169825624084040071190781216, −2.22636124926232743758354934975, −0.835671766477589081285097403245, 1.74592835914342932007214026742, 2.68414846342480031005116809245, 4.04304557074583223127565870918, 4.47602736197199566759139612991, 5.57817122955142603615790371469, 6.90253930288559556773994584391, 7.49026840281836881904154338719, 8.421597578500209812201930782649, 9.241072626593301741285670619365, 9.768569172411640397108191140440

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