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Properties

Label 2-1200-15.14-c2-0-23
Degree $2$
Conductor $1200$
Sign $0.867 - 0.498i$
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  + (1.65 − 2.5i)3-s + (−3.5 − 8.29i)9-s + 16.5i·11-s + 10i·13-s + 3.31·17-s + 7·19-s + 19.8·23-s + (−26.5 − 4.99i)27-s + 33.1i·29-s − 42·31-s + (41.4 + 27.5i)33-s + 40i·37-s + (25 + 16.5i)39-s + 16.5i·41-s + 50i·43-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)3-s + (−0.388 − 0.921i)9-s + 1.50i·11-s + 0.769i·13-s + 0.195·17-s + 0.368·19-s + 0.865·23-s + (−0.982 − 0.185i)27-s + 1.14i·29-s − 1.35·31-s + (1.25 + 0.833i)33-s + 1.08i·37-s + (0.641 + 0.425i)39-s + 0.404i·41-s + 1.16i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.867 - 0.498i$
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: \((2,\ 1200,\ (\ :1),\ 0.867 - 0.498i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.087469953\)
\(L(\frac12)\) \(\approx\) \(2.087469953\)
\(L(2)\) 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 \( 1 + (-1.65 + 2.5i)T \)
5 \( 1 \)
good7 \( 1 - 49T^{2} \)
11 \( 1 - 16.5iT - 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 - 3.31T + 289T^{2} \)
19 \( 1 - 7T + 361T^{2} \)
23 \( 1 - 19.8T + 529T^{2} \)
29 \( 1 - 33.1iT - 841T^{2} \)
31 \( 1 + 42T + 961T^{2} \)
37 \( 1 - 40iT - 1.36e3T^{2} \)
41 \( 1 - 16.5iT - 1.68e3T^{2} \)
43 \( 1 - 50iT - 1.84e3T^{2} \)
47 \( 1 - 46.4T + 2.20e3T^{2} \)
53 \( 1 - 46.4T + 2.80e3T^{2} \)
59 \( 1 + 66.3iT - 3.48e3T^{2} \)
61 \( 1 + 8T + 3.72e3T^{2} \)
67 \( 1 - 45iT - 4.48e3T^{2} \)
71 \( 1 - 33.1iT - 5.04e3T^{2} \)
73 \( 1 + 35iT - 5.32e3T^{2} \)
79 \( 1 - 12T + 6.24e3T^{2} \)
83 \( 1 - 69.6T + 6.88e3T^{2} \)
89 \( 1 + 149. iT - 7.92e3T^{2} \)
97 \( 1 - 70iT - 9.40e3T^{2} \)
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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

−9.373052515618442482022386103537, −8.930899241814704657334477729457, −7.82216262664159797992859467378, −7.14471312739781128472553650424, −6.64263594787134029475049971082, −5.40598260020880151674722326711, −4.39608834725759111131032366875, −3.27548720131382942772993140707, −2.17630845650918623954691431112, −1.26231668035046381412419823359, 0.61165414239177065140638244252, 2.40962405508497959720157168687, 3.35871393476091376541860979664, 4.04433729627616871594220781514, 5.42198162734690999409322330562, 5.73151238297019246861626465803, 7.22417924562641137320448634050, 8.001458940478116451323889809280, 8.836370534127625977030933714925, 9.293339947800426786857919767391

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