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Properties

Label 2-48e2-4.3-c0-0-4
Degree $2$
Conductor $2304$
Sign $i$
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  − 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

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048109754\)
\(L(\frac12)\) \(\approx\) \(1.048109754\)
\(L(1)\) 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

−9.114242497207640326853302073919, −7.81335997381449367189451315300, −7.70514338329259756358364289154, −6.73132206184206876853928528362, −6.01214017590718185201519041419, −4.83681416890478652245565010815, −4.07557428856026976188450948138, −3.47069462541115542879106263414, −2.03629666064916912360323311766, −0.71740962692815511497745289892, 1.75327063596125388044123665293, 2.63076695550441020484391687082, 3.52023894541209742038444549212, 4.82370201796253144795270162072, 5.47762330806930023468700620138, 6.15390944110622251810659672991, 6.98032801402658866611908423751, 8.108684525305321537858461185649, 8.599171721563966642211198950147, 9.309731868951967338933560600085

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