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Properties

Label 2-24e2-8.3-c0-0-1
Degree $2$
Conductor $576$
Sign $0.707 + 0.707i$
Analytic cond. $0.287461$
Root an. cond. $0.536154$
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  − 2i·7-s + 25-s + 2i·31-s − 3·49-s − 2·73-s + 2i·79-s + 2·97-s + 2i·103-s + ⋯
L(s)  = 1  − 2i·7-s + 25-s + 2i·31-s − 3·49-s − 2·73-s + 2i·79-s + 2·97-s + 2i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.287461\)
Root analytic conductor: \(0.536154\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8994119502\)
\(L(\frac12)\) \(\approx\) \(0.8994119502\)
\(L(1)\) 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 \( 1 - T^{2} \)
7 \( 1 + 2iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.57198892904563266978160118253, −10.28782934206358540109256084130, −9.105732270257483677005508464382, −8.062323081183420962365443700777, −7.16072820140762034799316090233, −6.59299598103872803363248794001, −5.08693297149565471584933806008, −4.17691721628773209667235471801, −3.17774740610349511358075071553, −1.23943299639996644350212698784, 2.09225422540142422006522603145, 3.07533934477735411260278544243, 4.62760404255712981340564764169, 5.64666139965654440650808722165, 6.28850164965973727119171079379, 7.59537091945565711854180860128, 8.607356024425571640643785833293, 9.146252725953161393245915333407, 10.05040523653622303018632606720, 11.26096362632694909936961615065

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