Properties

Label 2-1200-12.11-c1-0-17
Degree $2$
Conductor $1200$
Sign $0.912 - 0.408i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 − 0.707i)3-s + 4.24i·7-s + (2.00 − 2.23i)9-s + (3 + 6.70i)21-s + 9.48·23-s + (1.58 − 4.94i)27-s + 8.94i·29-s − 4.47i·41-s + 12.7i·43-s + 9.48·47-s − 10.9·49-s − 8·61-s + (9.48 + 8.48i)63-s − 4.24i·67-s + (15.0 − 6.70i)69-s + ⋯
L(s)  = 1  + (0.912 − 0.408i)3-s + 1.60i·7-s + (0.666 − 0.745i)9-s + (0.654 + 1.46i)21-s + 1.97·23-s + (0.304 − 0.952i)27-s + 1.66i·29-s − 0.698i·41-s + 1.94i·43-s + 1.38·47-s − 1.57·49-s − 1.02·61-s + (1.19 + 1.06i)63-s − 0.518i·67-s + (1.80 − 0.807i)69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.912 - 0.408i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331005848\)
\(L(\frac12)\) \(\approx\) \(2.331005848\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 9.48T + 23T^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 9.48T + 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.301250554805345830268443050309, −9.073521363571290174084166347157, −8.348986782664955408265497712229, −7.39194305892170427854402988491, −6.58856874111609122198008897354, −5.60907769289998755106605175634, −4.69008387541025363264653674050, −3.24408181734925928681211336652, −2.64654296774814580417924265032, −1.46242598362116845334824100896, 1.02500644120511307670571005275, 2.52409864054875588747354988307, 3.63487922013968776080021748011, 4.27312406216852418573863544590, 5.19478310939508148058502465999, 6.65435438878396866518088488243, 7.36787595225732594135510627652, 7.978169787560015087918798535045, 8.978087256848642350971043431372, 9.642591262140849168317303512474

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