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Properties

Label 2-1200-20.3-c1-0-11
Degree $2$
Conductor $1200$
Sign $0.287 + 0.957i$
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 + (−0.317 − 0.317i)7-s − 1.00i·9-s + 1.41i·11-s + (−2.44 − 2.44i)13-s + (0.449 − 0.449i)17-s + 6.29·19-s − 0.449·21-s + (4.87 − 4.87i)23-s + (−0.707 − 0.707i)27-s − 5.34i·29-s − 8.48i·31-s + (1.00 + 1.00i)33-s + (6.44 − 6.44i)37-s − 3.46·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.120 − 0.120i)7-s − 0.333i·9-s + 0.426i·11-s + (−0.679 − 0.679i)13-s + (0.109 − 0.109i)17-s + 1.44·19-s − 0.0980·21-s + (1.01 − 1.01i)23-s + (−0.136 − 0.136i)27-s − 0.993i·29-s − 1.52i·31-s + (0.174 + 0.174i)33-s + (1.06 − 1.06i)37-s − 0.554·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.287 + 0.957i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744934154\)
\(L(\frac12)\) \(\approx\) \(1.744934154\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.317 + 0.317i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + (-0.449 + 0.449i)T - 17iT^{2} \)
19 \( 1 - 6.29T + 19T^{2} \)
23 \( 1 + (-4.87 + 4.87i)T - 23iT^{2} \)
29 \( 1 + 5.34iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + (-6.44 + 6.44i)T - 37iT^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + (6.29 - 6.29i)T - 43iT^{2} \)
47 \( 1 + (-8.34 - 8.34i)T + 47iT^{2} \)
53 \( 1 + (-2 - 2i)T + 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + (0.635 + 0.635i)T + 67iT^{2} \)
71 \( 1 - 1.27iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 4.09T + 79T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−9.664355500137757125583483331457, −8.759058193431975452794489274661, −7.73503070335832828597592467063, −7.35809317135102829187349149158, −6.32002681115218077830990806643, −5.34411242112216665434925396176, −4.37328885276168657606597844748, −3.16476321483864066507367583546, −2.31610137272947042575627403992, −0.76433241838900594615926717685, 1.44818113184569169066888519693, 2.92448095476660997451506434727, 3.58507638824445805800297352442, 4.91253770605166193271765216563, 5.43161981710876407059890322894, 6.80965203926218182764093151442, 7.36786283356597563885076230525, 8.490213110990237389658376292319, 9.085724186547798994926048464478, 9.855391797556617893748252015100

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