Properties

Label 2-48e2-8.5-c1-0-7
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  − 2·7-s + 4i·11-s − 6i·13-s − 6·17-s + 4·23-s + 5·25-s + 4i·29-s + 10·31-s + 2i·37-s − 2·41-s + 8i·43-s + 12·47-s − 3·49-s + 12i·53-s + 4i·59-s + ⋯
L(s)  = 1  − 0.755·7-s + 1.20i·11-s − 1.66i·13-s − 1.45·17-s + 0.834·23-s + 25-s + 0.742i·29-s + 1.79·31-s + 0.328i·37-s − 0.312·41-s + 1.21i·43-s + 1.75·47-s − 0.428·49-s + 1.64i·53-s + 0.520i·59-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

\(L(1)\) \(\approx\) \(1.384875930\)
\(L(\frac12)\) \(\approx\) \(1.384875930\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.110811305845193024187413671908, −8.395300559329333244539484924573, −7.46618962569058683889751374971, −6.79656106667538285156952172592, −6.10083053128858053007161866083, −5.00550636684972483533667264911, −4.42424876166740720498172238122, −3.13160838501214896996928042036, −2.53484838795304958687526196288, −0.972974583802867190111204081286, 0.58923055414970614303377240373, 2.13107839483728461906007571724, 3.07563218131785714723678892280, 4.06396230468058290583962009579, 4.81069062198528587242616472805, 5.97295710103875823654212043959, 6.64029392073483037905432224881, 7.07818237263446246175161661843, 8.467787197629627861191951810615, 8.818442447272362450867433742920

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