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Properties

Label 2-2368-148.147-c0-0-0
Degree $2$
Conductor $2368$
Sign $1$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  − 1.73i·3-s + 1.73i·7-s − 1.99·9-s + 1.73i·11-s + 2.99·21-s + 25-s + 1.73i·27-s + 2.99·33-s + 37-s + 41-s + 1.73i·47-s − 1.99·49-s − 53-s − 3.46i·63-s − 1.73i·71-s + ⋯
L(s)  = 1  − 1.73i·3-s + 1.73i·7-s − 1.99·9-s + 1.73i·11-s + 2.99·21-s + 25-s + 1.73i·27-s + 2.99·33-s + 37-s + 41-s + 1.73i·47-s − 1.99·49-s − 53-s − 3.46i·63-s − 1.73i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $1$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (2367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087671447\)
\(L(\frac12)\) \(\approx\) \(1.087671447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 - T \)
good3 \( 1 + 1.73iT - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
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

−9.163624292699498090621066515470, −8.131295299173842170384293158511, −7.70057809234086842023635370639, −6.78800415253835358858762010798, −6.26109889425609610462854014511, −5.44697350517500104266864361522, −4.58429700926321065285739997679, −2.84582421369879677083942421569, −2.30827044410402804011478442289, −1.46124396291682490439625216753, 0.78508111203247531440531445542, 2.94487597444530737984696395659, 3.63668538904622061172498963309, 4.21504248118214788918653621125, 5.01736532782025949085912310481, 5.87221609200375922492376725239, 6.76679494396556988287041100509, 7.82736739647403873484907137699, 8.558108252258844648473568049855, 9.253745414595229766746336443743

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