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Properties

Label 2-2448-204.23-c0-0-0
Degree $2$
Conductor $2448$
Sign $-0.732 - 0.680i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.38 − 0.923i)5-s + (−1.30 + 1.30i)13-s − i·17-s + (0.675 + 1.63i)25-s + (−0.216 + 0.324i)29-s + (−1.63 + 0.324i)37-s + (−1.63 + 1.08i)41-s + (−0.382 + 0.923i)49-s + (−0.707 + 1.70i)53-s + (0.324 − 0.216i)61-s + (3.01 − 0.599i)65-s + (0.923 − 1.38i)73-s + (−0.923 + 1.38i)85-s + (−1.30 + 1.30i)89-s + (−1.63 − 1.08i)97-s + ⋯
L(s)  = 1  + (−1.38 − 0.923i)5-s + (−1.30 + 1.30i)13-s − i·17-s + (0.675 + 1.63i)25-s + (−0.216 + 0.324i)29-s + (−1.63 + 0.324i)37-s + (−1.63 + 1.08i)41-s + (−0.382 + 0.923i)49-s + (−0.707 + 1.70i)53-s + (0.324 − 0.216i)61-s + (3.01 − 0.599i)65-s + (0.923 − 1.38i)73-s + (−0.923 + 1.38i)85-s + (−1.30 + 1.30i)89-s + (−1.63 − 1.08i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-0.732 - 0.680i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ -0.732 - 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1523145170\)
\(L(\frac12)\) \(\approx\) \(0.1523145170\)
\(L(1)\) 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 \( 1 \)
17 \( 1 + iT \)
good5 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
97 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
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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

−9.254475651269268510557040675995, −8.702078555622992152181795607616, −7.82869911400717103936599469208, −7.26339394722651399872155594717, −6.56021894577842099366304924894, −5.06187826414242912936757561517, −4.78643836157909354837313847787, −3.92330651900862085075197745733, −2.91737389076073088576593675459, −1.54334484609267466648541933268, 0.099398250264513849623277329274, 2.15046191589007758311025787261, 3.27558296545927324182108383508, 3.72698699542296750018654110980, 4.85425151861255572501338712162, 5.65311590515507435879079187110, 6.89870800187634058880895530142, 7.18009305802347744243886937180, 8.175046094019620962255144923070, 8.417579719341213577985954290007

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