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Properties

Label 2-600-40.27-c1-0-13
Degree $2$
Conductor $600$
Sign $0.944 - 0.328i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
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  + (−1.10 + 0.887i)2-s + (0.707 − 0.707i)3-s + (0.423 − 1.95i)4-s + (−0.150 + 1.40i)6-s + (−1.17 + 1.17i)7-s + (1.26 + 2.52i)8-s − 1.00i·9-s + 4.03·11-s + (−1.08 − 1.68i)12-s + (0.571 + 0.571i)13-s + (0.249 − 2.32i)14-s + (−3.64 − 1.65i)16-s + (0.155 + 0.155i)17-s + (0.887 + 1.10i)18-s − 0.0474i·19-s + ⋯
L(s)  = 1  + (−0.778 + 0.627i)2-s + (0.408 − 0.408i)3-s + (0.211 − 0.977i)4-s + (−0.0614 + 0.574i)6-s + (−0.442 + 0.442i)7-s + (0.448 + 0.893i)8-s − 0.333i·9-s + 1.21·11-s + (−0.312 − 0.485i)12-s + (0.158 + 0.158i)13-s + (0.0666 − 0.622i)14-s + (−0.910 − 0.413i)16-s + (0.0378 + 0.0378i)17-s + (0.209 + 0.259i)18-s − 0.0108i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18409 + 0.199747i\)
\(L(\frac12)\) \(\approx\) \(1.18409 + 0.199747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.887i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (1.17 - 1.17i)T - 7iT^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + (-0.571 - 0.571i)T + 13iT^{2} \)
17 \( 1 + (-0.155 - 0.155i)T + 17iT^{2} \)
19 \( 1 + 0.0474iT - 19T^{2} \)
23 \( 1 + (-4.23 - 4.23i)T + 23iT^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + (5.25 - 5.25i)T - 37iT^{2} \)
41 \( 1 - 8.10T + 41T^{2} \)
43 \( 1 + (-7.00 + 7.00i)T - 43iT^{2} \)
47 \( 1 + (-5.25 + 5.25i)T - 47iT^{2} \)
53 \( 1 + (4.99 + 4.99i)T + 53iT^{2} \)
59 \( 1 - 9.81iT - 59T^{2} \)
61 \( 1 - 11.0iT - 61T^{2} \)
67 \( 1 + (-2.26 - 2.26i)T + 67iT^{2} \)
71 \( 1 + 5.87iT - 71T^{2} \)
73 \( 1 + (-8.04 + 8.04i)T - 73iT^{2} \)
79 \( 1 + 6.62T + 79T^{2} \)
83 \( 1 + (4.29 - 4.29i)T - 83iT^{2} \)
89 \( 1 - 11.6iT - 89T^{2} \)
97 \( 1 + (-1.35 - 1.35i)T + 97iT^{2} \)
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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

−10.49976768398700704463069429302, −9.371582827735229287965376976617, −9.099141064094599604767249884406, −8.108374179358415934336805303702, −7.15665448445533391344112550860, −6.42961046990794875787475983633, −5.59330057035095201754020959350, −4.10565670263292095320714552350, −2.57592836790751781182278569693, −1.15279779348930164041182166451, 1.13301420240523658442077677507, 2.76654367904061415889880436622, 3.69377694380510470694670305584, 4.65355923040436830896926841983, 6.43876579180564881138796082302, 7.16239206642837164363360172139, 8.320524559627279094555315643529, 9.017256272876094650122836656336, 9.678749069713644511523062314956, 10.60637715524723690253388313241

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