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Properties

Label 2-48e2-144.43-c0-0-1
Degree $2$
Conductor $2304$
Sign $0.461 + 0.887i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (−0.965 + 0.258i)3-s + (0.366 − 1.36i)5-s + (0.707 − 1.22i)7-s + (0.866 − 0.499i)9-s + (0.258 + 0.965i)11-s + 1.41i·15-s + 17-s + (0.707 + 0.707i)19-s + (−0.366 + 1.36i)21-s + (−0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (−0.366 − 1.36i)29-s + (−0.499 − 0.866i)33-s + (−1.41 − 1.41i)35-s + (1 + i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.366 − 1.36i)5-s + (0.707 − 1.22i)7-s + (0.866 − 0.499i)9-s + (0.258 + 0.965i)11-s + 1.41i·15-s + 17-s + (0.707 + 0.707i)19-s + (−0.366 + 1.36i)21-s + (−0.866 − 0.5i)25-s + (−0.707 + 0.707i)27-s + (−0.366 − 1.36i)29-s + (−0.499 − 0.866i)33-s + (−1.41 − 1.41i)35-s + (1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.461 + 0.887i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.461 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074735171\)
\(L(\frac12)\) \(\approx\) \(1.074735171\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
good5 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.385100919787821184858163332855, −8.068178263105037225858222916930, −7.64827047773057431034360659973, −6.67205989590685040607853749811, −5.72396471897375264896517408616, −4.98312239219850749144619041861, −4.47988409487558606603408510483, −3.71353467665407294982606802555, −1.65941138239380331139997074751, −1.00100997047126805795129588852, 1.44331580217306979377544966348, 2.59874711342526708926190306811, 3.40441020196032314444338729719, 4.84097372557487795524032730607, 5.68386230482923980228189498185, 5.99116674030672082255701913246, 6.96224696112110133190444670768, 7.55972786055636567541013490616, 8.542234623865950568260111588149, 9.395391578567103351239360518869

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