Properties

Label 2-48e2-8.5-c1-0-0
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 − 4i·11-s + 2i·13-s − 2·17-s + 4i·19-s − 8·23-s + 25-s + 6i·29-s + 8·31-s + 6i·37-s − 6·41-s + 4i·43-s − 7·49-s + 2i·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.894i·5-s − 1.20i·11-s + 0.554i·13-s − 0.485·17-s + 0.917i·19-s − 1.66·23-s + 0.200·25-s + 1.11i·29-s + 1.43·31-s + 0.986i·37-s − 0.937·41-s + 0.609i·43-s − 49-s + 0.274i·53-s + 1.07·55-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

\(L(1)\) \(\approx\) \(0.9732684211\)
\(L(\frac12)\) \(\approx\) \(0.9732684211\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−9.293519915764770492117794630436, −8.356040830883049492056128192654, −7.938371580462713834192878948760, −6.68218875941622221512850966157, −6.42217389574043078852694510307, −5.48503749812980556220380432607, −4.37885171867923715420089821869, −3.46762236550596602135600270505, −2.71995460686128039692108259734, −1.47608256025237012465449908004, 0.32496190451294687638815284237, 1.74000517821645087039298278051, 2.69099411753690930833341423639, 4.09953842333755903220408327640, 4.63388429120213377479475858963, 5.45088594781412998577898714603, 6.39646513753830288613233671949, 7.20838689010361996648963147661, 8.094825131228622968700602592412, 8.607220065156048829403997413640

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