Properties

Label 2-48e2-144.115-c0-0-2
Degree $2$
Conductor $2304$
Sign $0.953 + 0.300i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.953 + 0.300i$
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: \((2,\ 2304,\ (\ :0),\ 0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9913330859\)
\(L(\frac12)\) \(\approx\) \(0.9913330859\)
\(L(1)\) 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

−9.093788900336968908153360916950, −8.259575829397810147392737738280, −7.61653722279494060975679177894, −7.38209424236448950712441071038, −5.76168038733358743829221864715, −4.79092920084560334509919178346, −4.39929408269194825350283338913, −3.63053056009307360320197710727, −2.68831454556611454287507962448, −0.77460166788853807489614845908, 1.24472928348990391198171011593, 2.71241113812846995856487903373, 3.09334713476777655467979853976, 4.40403391580102678992032547056, 5.47275407105679898859132467891, 6.05457429406912584562754578183, 7.22552046494091747072290677355, 7.85061233590423849708451210549, 8.165032845985163807867596521387, 8.856814026539743196501812317443

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