Properties

Label 2-1200-15.14-c2-0-32
Degree $2$
Conductor $1200$
Sign $0.694 - 0.719i$
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  + (2.82 − i)3-s + 7i·7-s + (7.00 − 5.65i)9-s − 8.48i·11-s + 25i·13-s + 25.4·17-s − 7·19-s + (7 + 19.7i)21-s − 25.4·23-s + (14.1 − 23.0i)27-s + 42.4i·29-s + 7·31-s + (−8.48 − 24i)33-s + 2i·37-s + (25 + 70.7i)39-s + ⋯
L(s)  = 1  + (0.942 − 0.333i)3-s + i·7-s + (0.777 − 0.628i)9-s − 0.771i·11-s + 1.92i·13-s + 1.49·17-s − 0.368·19-s + (0.333 + 0.942i)21-s − 1.10·23-s + (0.523 − 0.851i)27-s + 1.46i·29-s + 0.225·31-s + (−0.257 − 0.727i)33-s + 0.0540i·37-s + (0.641 + 1.81i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.694 - 0.719i$
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: \((2,\ 1200,\ (\ :1),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.752909990\)
\(L(\frac12)\) \(\approx\) \(2.752909990\)
\(L(2)\) 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 \( 1 + (-2.82 + i)T \)
5 \( 1 \)
good7 \( 1 - 7iT - 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 - 25iT - 169T^{2} \)
17 \( 1 - 25.4T + 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 + 25.4T + 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 - 7T + 961T^{2} \)
37 \( 1 - 2iT - 1.36e3T^{2} \)
41 \( 1 + 8.48iT - 1.68e3T^{2} \)
43 \( 1 - 41iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 59.3T + 2.80e3T^{2} \)
59 \( 1 - 33.9iT - 3.48e3T^{2} \)
61 \( 1 + T + 3.72e3T^{2} \)
67 \( 1 + 17iT - 4.48e3T^{2} \)
71 \( 1 - 42.4iT - 5.04e3T^{2} \)
73 \( 1 - 70iT - 5.32e3T^{2} \)
79 \( 1 + 58T + 6.24e3T^{2} \)
83 \( 1 - 118.T + 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + 49iT - 9.40e3T^{2} \)
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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

−9.424166377990727530241271134791, −8.799878682627010753915350454446, −8.233742991471500324935315339049, −7.24786952364835124440880722676, −6.40411264189723278383452874385, −5.55600006775093966164655338703, −4.27732212041454734113724923231, −3.36327907802814100333814439448, −2.34940723502259828140793591411, −1.37836683145738001812477155565, 0.76144653249369402782094852072, 2.19666061526105055707763400553, 3.33440985171455197597220348942, 4.02391856487902037429145933374, 5.03854970225110971379856822461, 6.06019561065432534927676230370, 7.47612019808530877251120358993, 7.70092893531156189118766607370, 8.475707010246377208937597575725, 9.700919391542012521260181776610

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