Properties

Label 2-48e2-24.11-c1-0-14
Degree $2$
Conductor $2304$
Sign $0.169 - 0.985i$
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  + 3.41·5-s + 4.82i·7-s + 2.82i·11-s + 2.82i·13-s − 5.41i·17-s + 5.65·19-s − 1.17·23-s + 6.65·25-s − 0.585·29-s + 3.17i·31-s + 16.4i·35-s − 3.65i·37-s + 2.58i·41-s − 9.65·43-s + 12.4·47-s + ⋯
L(s)  = 1  + 1.52·5-s + 1.82i·7-s + 0.852i·11-s + 0.784i·13-s − 1.31i·17-s + 1.29·19-s − 0.244·23-s + 1.33·25-s − 0.108·29-s + 0.569i·31-s + 2.78i·35-s − 0.601i·37-s + 0.403i·41-s − 1.47·43-s + 1.82·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380635137\)
\(L(\frac12)\) \(\approx\) \(2.380635137\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 - 4.82iT - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 5.41iT - 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 + 0.585T + 29T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 5.07T + 53T^{2} \)
59 \( 1 + 2.34iT - 59T^{2} \)
61 \( 1 - 7.65iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 6.48iT - 79T^{2} \)
83 \( 1 + 5.17iT - 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−9.277155654074965225491697640459, −8.792287002570371706246089758869, −7.54725142908868946164707095591, −6.72406615080626481217633553512, −5.91608311452727762746857227518, −5.32822658451853229592401122351, −4.73052820495714928541401268023, −3.05166164318882793353193493911, −2.32251120395997807349617784552, −1.61123607492311535459949866335, 0.826585340299214730326717437238, 1.70625626132795416293482729059, 3.09378886361479788636458209660, 3.83748474723901262805046475903, 4.95590953664081951571103748715, 5.82469466625548586968915220488, 6.34705804031500724659682194126, 7.31770245710811620837178085465, 7.983687297878934146064798741805, 8.894288836591913855677111017774

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