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Properties

Label 2-48e2-3.2-c0-0-2
Degree $2$
Conductor $2304$
Sign $0.577 + 0.816i$
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  − 1.41i·5-s + 2·13-s + 1.41i·17-s − 1.00·25-s − 1.41i·29-s − 1.41i·41-s − 49-s − 1.41i·53-s − 2.82i·65-s + 2.00·85-s + 1.41i·89-s + 1.41i·101-s − 2·109-s + 1.41i·113-s + ⋯
L(s)  = 1  − 1.41i·5-s + 2·13-s + 1.41i·17-s − 1.00·25-s − 1.41i·29-s − 1.41i·41-s − 49-s − 1.41i·53-s − 2.82i·65-s + 2.00·85-s + 1.41i·89-s + 1.41i·101-s − 2·109-s + 1.41i·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279293523\)
\(L(\frac12)\) \(\approx\) \(1.279293523\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.866199880484436165541522623948, −8.401087997247697984858386715890, −7.891807497929631606741550366203, −6.52427500080091962280163570450, −5.93070951251310499423893168321, −5.16106570111755115839907576399, −4.11307933817956655566438453923, −3.63998345654676481895858272589, −1.96379194235026489583595942761, −1.03453254914688947298202793986, 1.44306342803683838422418478670, 2.89451592718283991994478146979, 3.30999418834642166141653840885, 4.40133938820781213414368472903, 5.53760756073587630381576624510, 6.36626167144009634914757567480, 6.88390947351416395135207169758, 7.66922307950815572033989082625, 8.551162696349199787057917049482, 9.309361098020824842874813663936

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