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Properties

Label 2-2400-120.107-c0-0-5
Degree $2$
Conductor $2400$
Sign $-0.559 + 0.828i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.559 + 0.828i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ -0.559 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082257880\)
\(L(\frac12)\) \(\approx\) \(1.082257880\)
\(L(1)\) 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

−8.850404177485007152454671763733, −8.135568887071482859096374610607, −7.41070661232255403704292592986, −6.46666547298134587162365203350, −6.03735049423147350261637380582, −5.08214758809088171957654050299, −3.79909814117354294672876238443, −3.00047351197616449221700554021, −2.02698487078208416048280700495, −0.68730169378393681016313771010, 1.87459035048984399193740352005, 2.81261766276377231908123865275, 3.91742498726915109474744118968, 4.60391347844255650433252267836, 5.23211089025911025170543640587, 6.29748985925294936339782605936, 7.19682827798984144199010617904, 7.950600132166587381448719577473, 8.764935228694953270829128413347, 9.612048868464700259795294949535

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