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Properties

Label 2-2400-480.323-c0-0-1
Degree $2$
Conductor $2400$
Sign $-0.936 + 0.349i$
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  + (−0.382 − 0.923i)2-s + (−0.382 − 0.923i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 0.707i)6-s + (0.923 + 0.382i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)12-s − i·16-s + (1.30 − 1.30i)17-s + (0.923 + 0.382i)18-s + (−0.292 + 0.707i)19-s − 0.765i·23-s − i·24-s + (0.923 + 0.382i)27-s − 1.41i·31-s + (−0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.382 − 0.923i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 0.707i)6-s + (0.923 + 0.382i)8-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)12-s − i·16-s + (1.30 − 1.30i)17-s + (0.923 + 0.382i)18-s + (−0.292 + 0.707i)19-s − 0.765i·23-s − i·24-s + (0.923 + 0.382i)27-s − 1.41i·31-s + (−0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.936 + 0.349i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ -0.936 + 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6960976469\)
\(L(\frac12)\) \(\approx\) \(0.6960976469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 \)
good7 \( 1 + T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
19 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + 0.765iT - T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
53 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
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

−8.733549411058835617384347463857, −8.070261362790465224162355520935, −7.47939648077898836443057349023, −6.63840434454986635838524136682, −5.57795742268729805289731033960, −4.87099352553319878899965455730, −3.68293451027635282155959161307, −2.72415438903296979684940830446, −1.83048600768335326199324407881, −0.62743478658030632897205649565, 1.33891888940821913997170961891, 3.20001569919281242022051882587, 4.07672550152726239520083227100, 4.95397038400185823921569500585, 5.61577051017303008857442992202, 6.30402828894670810580364474247, 7.12108787470916905546650831829, 8.138374061491019172232980349287, 8.618574270971896147569203374570, 9.557171606633066387894582735334

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