Properties

Label 2-1200-5.2-c2-0-35
Degree $2$
Conductor $1200$
Sign $-0.991 - 0.130i$
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 + (−6.12 − 6.12i)7-s − 2.99i·9-s − 6·11-s + (3.67 − 3.67i)13-s + (17.1 + 17.1i)17-s − 23i·19-s − 14.9·21-s + (12.2 − 12.2i)23-s + (−3.67 − 3.67i)27-s + 6i·29-s − 25·31-s + (−7.34 + 7.34i)33-s + (−24.4 − 24.4i)37-s − 9i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.874 − 0.874i)7-s − 0.333i·9-s − 0.545·11-s + (0.282 − 0.282i)13-s + (1.00 + 1.00i)17-s − 1.21i·19-s − 0.714·21-s + (0.532 − 0.532i)23-s + (−0.136 − 0.136i)27-s + 0.206i·29-s − 0.806·31-s + (−0.222 + 0.222i)33-s + (−0.662 − 0.662i)37-s − 0.230i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.991 - 0.130i$
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: \((2,\ 1200,\ (\ :1),\ -0.991 - 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6572719221\)
\(L(\frac12)\) \(\approx\) \(0.6572719221\)
\(L(2)\) 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 \( 1 + (6.12 + 6.12i)T + 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 169iT^{2} \)
17 \( 1 + (-17.1 - 17.1i)T + 289iT^{2} \)
19 \( 1 + 23iT - 361T^{2} \)
23 \( 1 + (-12.2 + 12.2i)T - 529iT^{2} \)
29 \( 1 - 6iT - 841T^{2} \)
31 \( 1 + 25T + 961T^{2} \)
37 \( 1 + (24.4 + 24.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 60T + 1.68e3T^{2} \)
43 \( 1 + (60.0 - 60.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (24.4 - 24.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 18iT - 3.48e3T^{2} \)
61 \( 1 + 37T + 3.72e3T^{2} \)
67 \( 1 + (-25.7 - 25.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 132T + 5.04e3T^{2} \)
73 \( 1 + (24.4 - 24.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 10iT - 6.24e3T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 6.88e3iT^{2} \)
89 \( 1 + 132iT - 7.92e3T^{2} \)
97 \( 1 + (-23.2 - 23.2i)T + 9.40e3iT^{2} \)
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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

−9.052208340705373389952907435480, −8.302469069692597076051958380256, −7.40393228709940096031556479125, −6.79741348571502496470554918852, −5.89934409882992625476250111462, −4.78298395808719744222346041661, −3.57118014050987901482664767492, −2.96562120442314982147003798158, −1.47687635041484351081835433437, −0.17788097427439186819908837285, 1.77247444014346842780790264651, 3.06236455875974967860217189171, 3.55617103678790265535894352613, 5.03577287820607549354276278763, 5.62609794392989111016584311681, 6.66113176017830216752410575509, 7.62095808663887868367533576299, 8.477501478718979251564964490705, 9.228594595779257878090430855686, 9.899738949784594596443353361063

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