Properties

Label 2-48e2-16.5-c1-0-0
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  + 0.378i·7-s + (−2.46 + 2.46i)13-s + (−2.44 + 2.44i)19-s − 5i·25-s − 10.1·31-s + (1.53 + 1.53i)37-s + (−7.34 − 7.34i)43-s + 6.85·49-s + (−10.4 + 10.4i)61-s + (−11.3 + 11.3i)67-s − 13.8i·73-s − 9.41·79-s + (−0.933 − 0.933i)91-s + 13.8·97-s + 11.6i·103-s + ⋯
L(s)  = 1  + 0.143i·7-s + (−0.683 + 0.683i)13-s + (−0.561 + 0.561i)19-s i·25-s − 1.82·31-s + (0.252 + 0.252i)37-s + (−1.12 − 1.12i)43-s + 0.979·49-s + (−1.33 + 1.33i)61-s + (−1.38 + 1.38i)67-s − 1.62i·73-s − 1.05·79-s + (−0.0978 − 0.0978i)91-s + 1.40·97-s + 1.15i·103-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: $\chi_{2304} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2261058370\)
\(L(\frac12)\) \(\approx\) \(0.2261058370\)
\(L(\frac{3}{2})\) 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 \( 1 + 5iT^{2} \)
7 \( 1 - 0.378iT - 7T^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + (2.46 - 2.46i)T - 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (2.44 - 2.44i)T - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + (-1.53 - 1.53i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.34 + 7.34i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (10.4 - 10.4i)T - 61iT^{2} \)
67 \( 1 + (11.3 - 11.3i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−9.197843585327449698378112847431, −8.764147730002048046160991426490, −7.76745872221598460646807678796, −7.10991523209489605594719789227, −6.26492720268980908782343801998, −5.47037131951553549638149302104, −4.53317229824779447960379692135, −3.76989677159229344038473931681, −2.58787161712220285390004411847, −1.67061549212658918453612669832, 0.07244231811309053405953560284, 1.62177224685869915673908173932, 2.77142668409070737687800926766, 3.66243766781013881886716977819, 4.70728173270438452221979721595, 5.41585950658977920837317491934, 6.29543954265114735027091037029, 7.25037574690372941158562362266, 7.71739118951589261216707981111, 8.718372190383889712243579614033

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