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Properties

Label 2-48e2-16.13-c1-0-22
Degree $2$
Conductor $2304$
Sign $0.991 + 0.130i$
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  + (−0.732 + 0.732i)5-s + 2.44i·7-s + (3.86 − 3.86i)11-s + (−3 − 3i)13-s + 3.46·17-s + (0.378 + 0.378i)19-s − 2.82i·23-s + 3.92i·25-s + (4.19 + 4.19i)29-s − 7.34·31-s + (−1.79 − 1.79i)35-s + (6.46 − 6.46i)37-s − 11.4i·41-s + (−2.44 + 2.44i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (−0.327 + 0.327i)5-s + 0.925i·7-s + (1.16 − 1.16i)11-s + (−0.832 − 0.832i)13-s + 0.840·17-s + (0.0869 + 0.0869i)19-s − 0.589i·23-s + 0.785i·25-s + (0.779 + 0.779i)29-s − 1.31·31-s + (−0.303 − 0.303i)35-s + (1.06 − 1.06i)37-s − 1.79i·41-s + (−0.373 + 0.373i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.716164331\)
\(L(\frac12)\) \(\approx\) \(1.716164331\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.732 - 0.732i)T - 5iT^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + (-3.86 + 3.86i)T - 11iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-0.378 - 0.378i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-4.19 - 4.19i)T + 29iT^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 + (-6.46 + 6.46i)T - 37iT^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + (2.44 - 2.44i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (-6.73 + 6.73i)T - 53iT^{2} \)
59 \( 1 + (-9.79 + 9.79i)T - 59iT^{2} \)
61 \( 1 + (-6.46 - 6.46i)T + 61iT^{2} \)
67 \( 1 + (0.757 + 0.757i)T + 67iT^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 + (-1.03 - 1.03i)T + 83iT^{2} \)
89 \( 1 - 8.92iT - 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−8.875648691115413394376343837715, −8.372240896204975358862417452616, −7.40710910810379386303325845787, −6.72888013622742522719441902636, −5.62325893077347087450774899609, −5.37338108225545085426458961965, −3.89514137868594529712967898241, −3.25994812932042074453498675862, −2.27400341725341084569231392796, −0.78178333465593547337805830339, 0.967947905072106939699776217539, 2.03419371975719671410154704716, 3.41283659961671777286069521757, 4.42036222191873323825300573926, 4.62375669835132640329821521339, 6.00108252051420449721192020644, 6.90288201299603072021190092223, 7.37774563190719765510088693818, 8.142654615937732145007974922720, 9.162975706523411496049231442738

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