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Properties

Label 2-48e2-48.11-c1-0-31
Degree $2$
Conductor $2304$
Sign $-0.220 + 0.975i$
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  + (2.57 − 2.57i)5-s + 3.74·7-s + (−3.64 − 3.64i)11-s + (−4.64 + 4.64i)13-s − 0.913i·17-s + (−1.41 − 1.41i)19-s − 6i·23-s − 8.29i·25-s + (−1.66 − 1.66i)29-s − 6.57i·31-s + (9.64 − 9.64i)35-s + (−1 − i)37-s − 0.913·41-s + (3.74 − 3.74i)43-s + 6·47-s + ⋯
L(s)  = 1  + (1.15 − 1.15i)5-s + 1.41·7-s + (−1.09 − 1.09i)11-s + (−1.28 + 1.28i)13-s − 0.221i·17-s + (−0.324 − 0.324i)19-s − 1.25i·23-s − 1.65i·25-s + (−0.309 − 0.309i)29-s − 1.18i·31-s + (1.63 − 1.63i)35-s + (−0.164 − 0.164i)37-s − 0.142·41-s + (0.570 − 0.570i)43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.022304485\)
\(L(\frac12)\) \(\approx\) \(2.022304485\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 + 2.57i)T - 5iT^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + (3.64 + 3.64i)T + 11iT^{2} \)
13 \( 1 + (4.64 - 4.64i)T - 13iT^{2} \)
17 \( 1 + 0.913iT - 17T^{2} \)
19 \( 1 + (1.41 + 1.41i)T + 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (1.66 + 1.66i)T + 29iT^{2} \)
31 \( 1 + 6.57iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 0.913T + 41T^{2} \)
43 \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (3.49 - 3.49i)T - 53iT^{2} \)
59 \( 1 + (-1.29 - 1.29i)T + 59iT^{2} \)
61 \( 1 + (-0.291 + 0.291i)T - 61iT^{2} \)
67 \( 1 + (3.32 + 3.32i)T + 67iT^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 8.58iT - 73T^{2} \)
79 \( 1 - 8.39iT - 79T^{2} \)
83 \( 1 + (4.93 - 4.93i)T - 83iT^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−8.848359610709895039932408526851, −8.142913931456640551169192162186, −7.41961290887867617711784029956, −6.24512296934969818220901408953, −5.46344730452389759881215186134, −4.85471221686446575088691927347, −4.32233427649618824712410618781, −2.45178195391633857090456911867, −1.99017875440674645688320945468, −0.63754524525460717533649850051, 1.69385847957087126049316656400, 2.35402857228229840295596833234, 3.20443046223421931147318411012, 4.74480844087056492374612637277, 5.26368119152462175845754143482, 5.93657803145602315244150105265, 7.23363908931494355673378084870, 7.46314907205754856921754489130, 8.280335855949084207745817611564, 9.438919917205187162814839918981

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