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Properties

Label 2-1200-20.3-c1-0-1
Degree $2$
Conductor $1200$
Sign $-0.727 - 0.685i$
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  + (−0.707 + 0.707i)3-s + (1.41 + 1.41i)7-s − 1.00i·9-s + 3.46i·11-s + (−2.44 − 2.44i)13-s + (−4.89 + 4.89i)17-s + 3.46·19-s − 2.00·21-s + (0.707 + 0.707i)27-s − 3.46i·31-s + (−2.44 − 2.44i)33-s + (−7.34 + 7.34i)37-s + 3.46·39-s + 6·41-s + (−5.65 + 5.65i)43-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.534 + 0.534i)7-s − 0.333i·9-s + 1.04i·11-s + (−0.679 − 0.679i)13-s + (−1.18 + 1.18i)17-s + 0.794·19-s − 0.436·21-s + (0.136 + 0.136i)27-s − 0.622i·31-s + (−0.426 − 0.426i)33-s + (−1.20 + 1.20i)37-s + 0.554·39-s + 0.937·41-s + (−0.862 + 0.862i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.727 - 0.685i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9262431871\)
\(L(\frac12)\) \(\approx\) \(0.9262431871\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + (4.89 - 4.89i)T - 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (7.34 - 7.34i)T - 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (-8.48 - 8.48i)T + 47iT^{2} \)
53 \( 1 + (4.89 + 4.89i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (4.89 + 4.89i)T + 73iT^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
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

−10.03604073099418675110808683093, −9.380997669047364522937946207886, −8.436535951711625198166262166877, −7.64496649553188476611820605897, −6.68807515242575720002280359900, −5.74626559798255167597554473913, −4.89990040920548701930635396857, −4.22718274053883988901120723633, −2.84703329607166560441864884220, −1.67025315090805819291695137102, 0.40923233487666378726184009493, 1.83861850757529903356156927381, 3.09730663063664735229148276110, 4.39490155929595856578136143513, 5.14382037364768310215446461807, 6.10869729736725106290472728437, 7.15839291211617450550139993856, 7.48465154255873382868996972914, 8.738491671608526474419618571363, 9.263726537417369112223616260025

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