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Properties

Label 2-1200-5.2-c2-0-23
Degree $2$
Conductor $1200$
Sign $-0.326 + 0.945i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + (−3.67 − 3.67i)7-s − 2.99i·9-s − 6·11-s + (6.12 − 6.12i)13-s + (22.0 + 22.0i)17-s + 25i·19-s + 9·21-s + (7.34 − 7.34i)23-s + (3.67 + 3.67i)27-s − 42i·29-s − 49·31-s + (7.34 − 7.34i)33-s + (4.89 + 4.89i)37-s + 14.9i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.524 − 0.524i)7-s − 0.333i·9-s − 0.545·11-s + (0.471 − 0.471i)13-s + (1.29 + 1.29i)17-s + 1.31i·19-s + 0.428·21-s + (0.319 − 0.319i)23-s + (0.136 + 0.136i)27-s − 1.44i·29-s − 1.58·31-s + (0.222 − 0.222i)33-s + (0.132 + 0.132i)37-s + 0.384i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6788203488\)
\(L(\frac12)\) \(\approx\) \(0.6788203488\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (3.67 + 3.67i)T + 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (-6.12 + 6.12i)T - 169iT^{2} \)
17 \( 1 + (-22.0 - 22.0i)T + 289iT^{2} \)
19 \( 1 - 25iT - 361T^{2} \)
23 \( 1 + (-7.34 + 7.34i)T - 529iT^{2} \)
29 \( 1 + 42iT - 841T^{2} \)
31 \( 1 + 49T + 961T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 1.36e3iT^{2} \)
41 \( 1 + 60T + 1.68e3T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 1.84e3iT^{2} \)
47 \( 1 + (51.4 + 51.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 78iT - 3.48e3T^{2} \)
61 \( 1 + 13T + 3.72e3T^{2} \)
67 \( 1 + (-52.6 - 52.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 60T + 5.04e3T^{2} \)
73 \( 1 + (-63.6 + 63.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 106iT - 6.24e3T^{2} \)
83 \( 1 + (-80.8 + 80.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 60iT - 7.92e3T^{2} \)
97 \( 1 + (121. + 121. i)T + 9.40e3iT^{2} \)
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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

−9.600987579577636772125546402006, −8.310800671997849032489396968245, −7.84727067127762692901748973598, −6.66309176233734348364999375297, −5.87113081221152322661267972712, −5.18806233982789407191048920131, −3.82355370038079837736667357781, −3.41026988846220617941485397927, −1.69142531376407168140476160628, −0.23028244290059556018683163379, 1.19519753946113291602599968571, 2.60820295195488558602547876954, 3.46994776856508752309053620140, 5.04047541695511024727871454620, 5.43084708407577714365117419896, 6.62151058570814266215166251512, 7.16860336998673250049256423030, 8.099525267310405960744139596492, 9.170917759442954622463352966489, 9.591141170524219895373420033383

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