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Properties

Label 2-1200-60.23-c0-0-4
Degree $2$
Conductor $1200$
Sign $-0.189 + 0.981i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
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 + (−0.707 − 0.707i)7-s − 1.00i·9-s + (−1.22 − 1.22i)13-s − 1.73·19-s + 1.00·21-s + (0.707 + 0.707i)27-s − 1.73i·31-s + 1.73·39-s + (0.707 − 0.707i)43-s + (1.22 − 1.22i)57-s − 61-s + (−0.707 + 0.707i)63-s + (0.707 + 0.707i)67-s − 1.00·81-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (−1.22 − 1.22i)13-s − 1.73·19-s + 1.00·21-s + (0.707 + 0.707i)27-s − 1.73i·31-s + 1.73·39-s + (0.707 − 0.707i)43-s + (1.22 − 1.22i)57-s − 61-s + (−0.707 + 0.707i)63-s + (0.707 + 0.707i)67-s − 1.00·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3997793779\)
\(L(\frac12)\) \(\approx\) \(0.3997793779\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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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

−9.943433583213664888760250732967, −9.158977486088443733946571229179, −8.043636083731500701590825307348, −7.12984549875468756413623845312, −6.28603028987181629862329901729, −5.49503268003706286822728789254, −4.48889629249691705884979550839, −3.75335578319457017535164966779, −2.55360387859411942940465678206, −0.35899072273799486367585526515, 1.84433059421072079853506284368, 2.74830163598140355564122869026, 4.33958209335588569819012305705, 5.14904439404301878634029332539, 6.25329938941199050081285913228, 6.66925931155751812537188691646, 7.53420922069594823484449656242, 8.597826089865494577776182979791, 9.308454451316691104724479295156, 10.24818741350516769620967862965

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