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Properties

Label 2-1200-5.2-c2-0-25
Degree $2$
Conductor $1200$
Sign $0.229 + 0.973i$
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 + (−1.44 − 1.44i)7-s − 2.99i·9-s + 3.34·11-s + (10.4 − 10.4i)13-s + (2.65 + 2.65i)17-s + 20.6i·19-s − 3.55·21-s + (16.4 − 16.4i)23-s + (−3.67 − 3.67i)27-s + 0.853i·29-s + 18.6·31-s + (4.10 − 4.10i)33-s + (−38.0 − 38.0i)37-s − 25.5i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.207 − 0.207i)7-s − 0.333i·9-s + 0.304·11-s + (0.803 − 0.803i)13-s + (0.155 + 0.155i)17-s + 1.08i·19-s − 0.169·21-s + (0.715 − 0.715i)23-s + (−0.136 − 0.136i)27-s + 0.0294i·29-s + 0.603·31-s + (0.124 − 0.124i)33-s + (−1.02 − 1.02i)37-s − 0.656i·39-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.192150592\)
\(L(\frac12)\) \(\approx\) \(2.192150592\)
\(L(2)\) 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 \( 1 + (1.44 + 1.44i)T + 49iT^{2} \)
11 \( 1 - 3.34T + 121T^{2} \)
13 \( 1 + (-10.4 + 10.4i)T - 169iT^{2} \)
17 \( 1 + (-2.65 - 2.65i)T + 289iT^{2} \)
19 \( 1 - 20.6iT - 361T^{2} \)
23 \( 1 + (-16.4 + 16.4i)T - 529iT^{2} \)
29 \( 1 - 0.853iT - 841T^{2} \)
31 \( 1 - 18.6T + 961T^{2} \)
37 \( 1 + (38.0 + 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.6T + 1.68e3T^{2} \)
43 \( 1 + (-22.4 + 22.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.7 - 19.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (28.6 - 28.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (54.8 + 54.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-39.7 + 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (21.1 - 21.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (14.5 + 14.5i)T + 9.40e3iT^{2} \)
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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

−9.272889803607304356820914896185, −8.470842606184181439411301567025, −7.86698072164209681344278696659, −6.88785304801776274094565408926, −6.15392577428109212029280803199, −5.20886421391440096425202650678, −3.89142086321102563736319360407, −3.20729464741117359727453362901, −1.89603427580325878357416799726, −0.68458432275619475398891340959, 1.24332589706096673458788854861, 2.60707029133905077735632970980, 3.55599230790551428682359355946, 4.49302057937257021683009906832, 5.41103080061491809403738222687, 6.51059468600811607625793337532, 7.17886805406602329574880333438, 8.337716807377456589103828645942, 8.967449468633943085363478447650, 9.567583615925942890044937316800

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