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Properties

Label 2-1200-3.2-c2-0-37
Degree $2$
Conductor $1200$
Sign $0.949 + 0.312i$
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  + (−2.84 − 0.938i)3-s + 6.81·7-s + (7.23 + 5.34i)9-s − 7.52i·11-s + 16.2·13-s + 4.11i·17-s + 7.86·19-s + (−19.4 − 6.39i)21-s − 19.5i·23-s + (−15.6 − 22.0i)27-s + 55.8i·29-s − 43.4·31-s + (−7.06 + 21.4i)33-s + 31.5·37-s + (−46.2 − 15.2i)39-s + ⋯
L(s)  = 1  + (−0.949 − 0.312i)3-s + 0.973·7-s + (0.804 + 0.594i)9-s − 0.684i·11-s + 1.24·13-s + 0.242i·17-s + 0.413·19-s + (−0.924 − 0.304i)21-s − 0.848i·23-s + (−0.578 − 0.815i)27-s + 1.92i·29-s − 1.40·31-s + (−0.214 + 0.650i)33-s + 0.853·37-s + (−1.18 − 0.390i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.949 + 0.312i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.709272330\)
\(L(\frac12)\) \(\approx\) \(1.709272330\)
\(L(2)\) 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 \( 1 + (2.84 + 0.938i)T \)
5 \( 1 \)
good7 \( 1 - 6.81T + 49T^{2} \)
11 \( 1 + 7.52iT - 121T^{2} \)
13 \( 1 - 16.2T + 169T^{2} \)
17 \( 1 - 4.11iT - 289T^{2} \)
19 \( 1 - 7.86T + 361T^{2} \)
23 \( 1 + 19.5iT - 529T^{2} \)
29 \( 1 - 55.8iT - 841T^{2} \)
31 \( 1 + 43.4T + 961T^{2} \)
37 \( 1 - 31.5T + 1.36e3T^{2} \)
41 \( 1 - 51.3iT - 1.68e3T^{2} \)
43 \( 1 - 51.2T + 1.84e3T^{2} \)
47 \( 1 - 61.7iT - 2.20e3T^{2} \)
53 \( 1 + 82.7iT - 2.80e3T^{2} \)
59 \( 1 + 97.6iT - 3.48e3T^{2} \)
61 \( 1 - 4.13T + 3.72e3T^{2} \)
67 \( 1 - 63.1T + 4.48e3T^{2} \)
71 \( 1 - 40.3iT - 5.04e3T^{2} \)
73 \( 1 - 78.5T + 5.32e3T^{2} \)
79 \( 1 + 51.0T + 6.24e3T^{2} \)
83 \( 1 - 2.72iT - 6.88e3T^{2} \)
89 \( 1 + 70.4iT - 7.92e3T^{2} \)
97 \( 1 + 3.44T + 9.40e3T^{2} \)
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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

−9.541534074873251404971895227935, −8.518432409443774401662562206367, −7.926842227802853852340459128917, −6.91622992515831147980360116218, −6.10326116364784804115133932948, −5.35611391398136129727284127255, −4.52175634397206992705863653721, −3.39254541779735770239184999166, −1.77557911550211113708435618641, −0.864574463470070689545897069593, 0.874106542613344827130321941155, 2.01108314710145762426130274247, 3.75450280055759976934988681340, 4.43727941246556574121871738267, 5.46421587463115504744021138995, 5.99753110476752159950946784548, 7.17996407552618926014437279607, 7.79019393698094659405298785202, 8.930720692918300807973015132011, 9.654777102146736584023015116291

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