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Properties

Label 2-2400-120.83-c0-0-1
Degree $2$
Conductor $2400$
Sign $0.923 - 0.382i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.866 + 0.499i)9-s + 1.73i·11-s + (0.707 + 0.707i)17-s + i·19-s + (0.707 + 0.707i)27-s + (1.67 − 0.448i)33-s + 1.73i·41-s − i·49-s + (0.500 − 0.866i)51-s + (0.965 − 0.258i)57-s + (1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s + (0.500 − 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.866 + 0.499i)9-s + 1.73i·11-s + (0.707 + 0.707i)17-s + i·19-s + (0.707 + 0.707i)27-s + (1.67 − 0.448i)33-s + 1.73i·41-s − i·49-s + (0.500 − 0.866i)51-s + (0.965 − 0.258i)57-s + (1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s + (0.500 − 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9837841645\)
\(L(\frac12)\) \(\approx\) \(0.9837841645\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.190298158538977996901692239652, −8.074146104563614812938723738823, −7.75757282662236714447159460637, −6.87611907826001377666957402361, −6.23865926289493646648032473090, −5.36000447310506801523251350229, −4.52455567240609679922249908986, −3.40077123692003507962726773793, −2.17493723178632031920049787667, −1.43504208371154466812001841293, 0.72984464960770401999361856084, 2.69703380465988015137939010187, 3.38627239398283981752270106147, 4.26407550247963211491541406035, 5.29485431883283806126914525487, 5.73441374658461554786096323773, 6.65402131308746304542760961555, 7.64265509446526429745478024923, 8.662954282247143403699133348525, 8.981141431853670685902597457995

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