Properties

Label 2-1200-15.14-c2-0-38
Degree $2$
Conductor $1200$
Sign $0.612 + 0.790i$
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  + (−2.94 + 0.581i)3-s + 11.4i·7-s + (8.32 − 3.42i)9-s − 8.48i·11-s − 10i·13-s − 3.55·17-s − 10.9·19-s + (−6.67 − 33.8i)21-s − 17.6·23-s + (−22.5 + 14.9i)27-s − 26.8i·29-s − 8·31-s + (4.93 + 24.9i)33-s + 59.9i·37-s + (5.81 + 29.4i)39-s + ⋯
L(s)  = 1  + (−0.981 + 0.193i)3-s + 1.64i·7-s + (0.924 − 0.380i)9-s − 0.771i·11-s − 0.769i·13-s − 0.209·17-s − 0.577·19-s + (−0.317 − 1.60i)21-s − 0.767·23-s + (−0.833 + 0.552i)27-s − 0.925i·29-s − 0.258·31-s + (0.149 + 0.756i)33-s + 1.62i·37-s + (0.149 + 0.754i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.612 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8643555807\)
\(L(\frac12)\) \(\approx\) \(0.8643555807\)
\(L(2)\) 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 \( 1 + (2.94 - 0.581i)T \)
5 \( 1 \)
good7 \( 1 - 11.4iT - 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 + 3.55T + 289T^{2} \)
19 \( 1 + 10.9T + 361T^{2} \)
23 \( 1 + 17.6T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 - 59.9iT - 1.36e3T^{2} \)
41 \( 1 - 20.5iT - 1.68e3T^{2} \)
43 \( 1 + 42.4iT - 1.84e3T^{2} \)
47 \( 1 - 88.2T + 2.20e3T^{2} \)
53 \( 1 - 3.55T + 2.80e3T^{2} \)
59 \( 1 + 77.7iT - 3.48e3T^{2} \)
61 \( 1 - 21.9T + 3.72e3T^{2} \)
67 \( 1 + 53.5iT - 4.48e3T^{2} \)
71 \( 1 + 69.2iT - 5.04e3T^{2} \)
73 \( 1 - 12.0iT - 5.32e3T^{2} \)
79 \( 1 + 9.02T + 6.24e3T^{2} \)
83 \( 1 + 0.688T + 6.88e3T^{2} \)
89 \( 1 + 7.10iT - 7.92e3T^{2} \)
97 \( 1 + 111. iT - 9.40e3T^{2} \)
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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

−9.498017786724275522190236093074, −8.634931246503768965610946087434, −7.944385652467487497163180895900, −6.62788876203397693361560956578, −5.89053533563478915492739729082, −5.46048502337782822946730447023, −4.41391403240051678862595289507, −3.16976294585930221819404727759, −1.99943413316431309783837867279, −0.36641356952153239996427159973, 0.933578979125606430085560115658, 2.06488933844797202257250314781, 4.02638322279851492857298008523, 4.30326856700036909941712595859, 5.45564580843613045226101976790, 6.50971840141920402070110917742, 7.16915795870692040188604446210, 7.63104447637744610990521583669, 8.977008474627605171611200355998, 9.938182066663411213346705511359

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