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Properties

Label 2-48e2-144.115-c0-0-3
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  + (−0.258 − 0.965i)3-s + (1.36 + 0.366i)5-s + (0.707 − 1.22i)7-s + (−0.866 + 0.499i)9-s + (0.965 − 0.258i)11-s − 1.41i·15-s + 17-s + (−0.707 + 0.707i)19-s + (−1.36 − 0.366i)21-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (−1.36 + 0.366i)29-s + (−0.499 − 0.866i)33-s + (1.41 − 1.41i)35-s + (−1 + i)37-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (1.36 + 0.366i)5-s + (0.707 − 1.22i)7-s + (−0.866 + 0.499i)9-s + (0.965 − 0.258i)11-s − 1.41i·15-s + 17-s + (−0.707 + 0.707i)19-s + (−1.36 − 0.366i)21-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (−1.36 + 0.366i)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: $\chi_{2304} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.461 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530282698\)
\(L(\frac12)\) \(\approx\) \(1.530282698\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
good5 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
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

−8.990166186502709072859379684477, −8.179978556231716398039996441200, −7.33571702682178460138790968998, −6.79135494007996555687081432600, −5.96393095034754982278023813517, −5.43696595518628547235118249864, −4.22400367601333588878729159920, −3.14534374283443612666647979976, −1.78882939876126266476104031113, −1.33571065506612210273183726822, 1.63203254886568762225341422251, 2.47935826150510691171717264638, 3.72112502659663581610900296904, 4.75800800071514333732384688940, 5.46158568952994062231464123842, 5.85579220303239332253771657987, 6.72998798876882909370849189683, 8.078140075943486167784894624421, 8.945142825154344328881072424729, 9.278711273708776289853708801118

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