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Properties

Label 2-48e2-16.3-c0-0-0
Degree $2$
Conductor $2304$
Sign $-0.923 - 0.382i$
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  − 1.41·7-s + (−1 − i)13-s + (−1.41 + 1.41i)19-s + i·25-s + 1.41i·31-s + (−1 + i)37-s + (−1.41 − 1.41i)43-s + 1.00·49-s + (−1 − i)61-s − 1.41i·79-s + (1.41 + 1.41i)91-s − 1.41·103-s + (1 + i)109-s + ⋯
L(s)  = 1  − 1.41·7-s + (−1 − i)13-s + (−1.41 + 1.41i)19-s + i·25-s + 1.41i·31-s + (−1 + i)37-s + (−1.41 − 1.41i)43-s + 1.00·49-s + (−1 − i)61-s − 1.41i·79-s + (1.41 + 1.41i)91-s − 1.41·103-s + (1 + i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1779741618\)
\(L(\frac12)\) \(\approx\) \(0.1779741618\)
\(L(1)\) 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 \( 1 - iT^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
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

−9.618292646854421786414355708991, −8.763689757247044861762314383832, −8.039296437814855261216236881972, −7.08987441126375181163693687388, −6.49753798095318609385682754914, −5.65482761314676675087998370801, −4.85130223293669523379985459791, −3.58299271966217620820163601233, −3.10820471681110622902501709595, −1.82891550943925839986625081560, 0.10988874159758579008508723890, 2.15809311583789176247912629723, 2.88873767217867023359194083431, 4.07469871901779688092241481604, 4.69494694504989562345847928257, 5.89552621973001947210285539648, 6.67835761935869480428379474560, 7.01679888325089466211047989219, 8.148143711572473881557204080634, 9.054633312872521515384438179266

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