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Properties

Label 2-48e2-12.11-c1-0-18
Degree $2$
Conductor $2304$
Sign $0.816 - 0.577i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44i·5-s + 2i·7-s + 4.89·11-s + 3.46·13-s − 4.24i·17-s − 6.92i·19-s + 8.48·23-s − 0.999·25-s − 7.34i·29-s + 2i·31-s − 4.89·35-s − 6.92·37-s + 4.24i·41-s + 8.48·47-s + 3·49-s + ⋯
L(s)  = 1  + 1.09i·5-s + 0.755i·7-s + 1.47·11-s + 0.960·13-s − 1.02i·17-s − 1.58i·19-s + 1.76·23-s − 0.199·25-s − 1.36i·29-s + 0.359i·31-s − 0.828·35-s − 1.13·37-s + 0.662i·41-s + 1.23·47-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.179475785\)
\(L(\frac12)\) \(\approx\) \(2.179475785\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.44iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 - 2.44iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 6.92iT - 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 - 4.89T + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−9.060890347334159817886986013207, −8.563991486783831119517733026214, −7.18176539073575559584774012033, −6.86385441155677302328006573426, −6.13067662653973925922595847906, −5.17039990309429690474302921785, −4.16534758217647892315002082240, −3.12338176835940770015394716436, −2.52948184473363321141289681488, −1.05208011942309155525426782425, 1.09065120926770820063246310275, 1.55694744166573482409835677213, 3.55367544704249605263066157122, 3.90753085374511662244100508989, 4.90101863950603248879441320812, 5.79823705499227704859683804460, 6.59509333004898273289634208473, 7.36851190314382552730185948702, 8.422181833849579040636338950338, 8.840423343673042325686516617854

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