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Properties

Label 2-1200-3.2-c2-0-43
Degree $2$
Conductor $1200$
Sign $0.596 + 0.802i$
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  + (−1.78 − 2.40i)3-s + 12.2·7-s + (−2.60 + 8.61i)9-s − 8.79i·11-s + 6.20·13-s − 16.6i·17-s + 8.93·19-s + (−21.8 − 29.3i)21-s + 40.1i·23-s + (25.4 − 9.14i)27-s + 4.94i·29-s + 50.6·31-s + (−21.1 + 15.7i)33-s − 41.3·37-s + (−11.0 − 14.9i)39-s + ⋯
L(s)  = 1  + (−0.596 − 0.802i)3-s + 1.74·7-s + (−0.288 + 0.957i)9-s − 0.799i·11-s + 0.477·13-s − 0.977i·17-s + 0.470·19-s + (−1.03 − 1.39i)21-s + 1.74i·23-s + (0.940 − 0.338i)27-s + 0.170i·29-s + 1.63·31-s + (−0.641 + 0.476i)33-s − 1.11·37-s + (−0.284 − 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.596 + 0.802i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.596 + 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.057224069\)
\(L(\frac12)\) \(\approx\) \(2.057224069\)
\(L(2)\) 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 \( 1 + (1.78 + 2.40i)T \)
5 \( 1 \)
good7 \( 1 - 12.2T + 49T^{2} \)
11 \( 1 + 8.79iT - 121T^{2} \)
13 \( 1 - 6.20T + 169T^{2} \)
17 \( 1 + 16.6iT - 289T^{2} \)
19 \( 1 - 8.93T + 361T^{2} \)
23 \( 1 - 40.1iT - 529T^{2} \)
29 \( 1 - 4.94iT - 841T^{2} \)
31 \( 1 - 50.6T + 961T^{2} \)
37 \( 1 + 41.3T + 1.36e3T^{2} \)
41 \( 1 + 10.8iT - 1.68e3T^{2} \)
43 \( 1 - 39.5T + 1.84e3T^{2} \)
47 \( 1 - 78.2iT - 2.20e3T^{2} \)
53 \( 1 - 69.7iT - 2.80e3T^{2} \)
59 \( 1 + 96.0iT - 3.48e3T^{2} \)
61 \( 1 - 26.7T + 3.72e3T^{2} \)
67 \( 1 - 66.5T + 4.48e3T^{2} \)
71 \( 1 + 34.7iT - 5.04e3T^{2} \)
73 \( 1 + 70.5T + 5.32e3T^{2} \)
79 \( 1 + 31.2T + 6.24e3T^{2} \)
83 \( 1 + 71.5iT - 6.88e3T^{2} \)
89 \( 1 + 30.4iT - 7.92e3T^{2} \)
97 \( 1 - 108.T + 9.40e3T^{2} \)
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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

−9.308516870504648960948362887315, −8.353278595108153539781435147163, −7.78873308228916605898692041190, −7.10459406767732301516975633218, −5.93304657996441161610105787774, −5.30585480902101517211729610310, −4.50588254527701924972234919892, −3.01582264683202887473598105657, −1.69652080228623216416513433243, −0.891537675073960800591590314377, 1.01997858812742255234590187432, 2.28361256190910956188045449556, 3.87775719617457583160151088740, 4.59282194768494776480642583346, 5.21821173650662255105409809942, 6.20777260221833564971688337544, 7.15806092320176716614693029986, 8.389628118407346079810480747251, 8.605413938588551032318502675980, 9.968931542223785919253952803310

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