Properties

Label 2-48e2-8.5-c1-0-23
Degree $2$
Conductor $2304$
Sign $0.707 + 0.707i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·5-s + 4·7-s + 2i·11-s − 2i·13-s + 2·17-s − 2i·19-s + 4·23-s + 25-s − 6i·29-s − 8i·35-s + 10i·37-s − 6·41-s + 6i·43-s + 8·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.894i·5-s + 1.51·7-s + 0.603i·11-s − 0.554i·13-s + 0.485·17-s − 0.458i·19-s + 0.834·23-s + 0.200·25-s − 1.11i·29-s − 1.35i·35-s + 1.64i·37-s − 0.937·41-s + 0.914i·43-s + 1.16·47-s + 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.243204935\)
\(L(\frac12)\) \(\approx\) \(2.243204935\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.782890792057012148717923048519, −8.117736686076816284090835377516, −7.64161367339961920027084258393, −6.62180298459201982165834510160, −5.48692317258635396260497518279, −4.85617985742319696959468116782, −4.41324315214423738789174059540, −3.03417653043187040154959811955, −1.81598920577219138278859064481, −0.923414068528069850584061886816, 1.22108427632985252910194514225, 2.28141504243485475984353723068, 3.32705409726734369332605831933, 4.24199522480338224179017852798, 5.23660894146425618050434861345, 5.84930303477066848876127724748, 7.12309300166512313565346938203, 7.29963258870782483710472540334, 8.509755686003725234299752697398, 8.793760956749197758015793851178

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