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Properties

Label 2-48e2-144.139-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: $\chi_{2304} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ -0.461 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.378698619\)
\(L(\frac12)\) \(\approx\) \(1.378698619\)
\(L(1)\) 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

−9.109874627166334200453538183172, −7.84728091984015357919308102740, −7.56152460272012701724120852076, −6.61938428445263842319655434774, −5.75349124112670527584040360531, −5.41387065151071605712477990419, −3.86084526024541435629952154289, −2.98234714535537299689968012706, −1.93894633505965744101513759932, −0.903677304998079378541352215973, 2.09592863091422994892390533093, 2.76807555302392370328873369225, 3.47676166089465420936375205296, 4.98246441753292638382805783847, 5.53581657857982792850153167913, 5.95561028839456247010938354721, 7.07796282976504235123348609683, 8.077846696725923559023027347175, 9.062047897854816263575195329893, 9.450551096678034717351644911733

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