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Properties

Label 2-2448-204.71-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.397 + 0.917i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415362403\)
\(L(\frac12)\) \(\approx\) \(1.415362403\)
\(L(1)\) 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

−9.147946165022182243828682945644, −8.313585733466033905969363383227, −7.47592130653436100120887573170, −6.64462843754459892455442263239, −5.49375268584721143322877933795, −5.33660479228023543855574955185, −4.43725765544705731314525481258, −2.96333252245768712825092046440, −2.22748431317440958143898850625, −0.950898855193738584967688489464, 1.85295189467096260531453549297, 2.32875396347045495618128895506, 3.47848261134448345466161897287, 4.57640420643956117200585489149, 5.48075852521328071096328892230, 6.27639204564341509657362271767, 6.84452298660414070224758069887, 7.53117137542074172200938785522, 8.693464037142857958270219757432, 9.405839989530210375686676964734

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