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Properties

Label 2-1200-5.2-c2-0-31
Degree $2$
Conductor $1200$
Sign $-0.973 + 0.229i$
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 + (−7.34 − 7.34i)7-s − 2.99i·9-s + 18·11-s + (−7.34 + 7.34i)13-s + (−4.89 − 4.89i)17-s + 10i·19-s + 18·21-s + (19.5 − 19.5i)23-s + (3.67 + 3.67i)27-s − 22·31-s + (−22.0 + 22.0i)33-s + (7.34 + 7.34i)37-s − 18i·39-s − 18·41-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−1.04 − 1.04i)7-s − 0.333i·9-s + 1.63·11-s + (−0.565 + 0.565i)13-s + (−0.288 − 0.288i)17-s + 0.526i·19-s + 0.857·21-s + (0.851 − 0.851i)23-s + (0.136 + 0.136i)27-s − 0.709·31-s + (−0.668 + 0.668i)33-s + (0.198 + 0.198i)37-s − 0.461i·39-s − 0.439·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
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: \((2,\ 1200,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1871890932\)
\(L(\frac12)\) \(\approx\) \(0.1871890932\)
\(L(2)\) 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 \( 1 + (7.34 + 7.34i)T + 49iT^{2} \)
11 \( 1 - 18T + 121T^{2} \)
13 \( 1 + (7.34 - 7.34i)T - 169iT^{2} \)
17 \( 1 + (4.89 + 4.89i)T + 289iT^{2} \)
19 \( 1 - 10iT - 361T^{2} \)
23 \( 1 + (-19.5 + 19.5i)T - 529iT^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 22T + 961T^{2} \)
37 \( 1 + (-7.34 - 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + (29.3 - 29.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.0 + 44.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-4.89 + 4.89i)T - 2.80e3iT^{2} \)
59 \( 1 - 90iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (44.0 + 44.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 72T + 5.04e3T^{2} \)
73 \( 1 + (44.0 - 44.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 70iT - 6.24e3T^{2} \)
83 \( 1 + (53.8 - 53.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 90iT - 7.92e3T^{2} \)
97 \( 1 + (102. + 102. i)T + 9.40e3iT^{2} \)
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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

−9.391686931924602978603418054265, −8.581742284083891657096502507129, −7.09260159855041882427425762231, −6.82338359782328825452409851045, −5.96263808819646318038187801723, −4.63935328473384988718734047012, −4.00359723737504247754444617890, −3.11196762625280652069811227666, −1.38793786638847632015330411662, −0.06137424429916819463613429294, 1.46954510523369084908743948981, 2.74821410303708808044934980317, 3.70086150080348084497032238001, 5.01060026893579634983812315356, 5.87107162135130488514114328935, 6.59263167882699103693159231967, 7.20984099846477587059580541124, 8.431477834843658442149919985252, 9.264130987178447321485181667818, 9.654195437632362809663059693763

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