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Properties

Label 2-48e2-24.11-c1-0-21
Degree $2$
Conductor $2304$
Sign $0.985 + 0.169i$
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  + 4.24·5-s − 4i·13-s + 4.24i·17-s + 12.9·25-s + 4.24·29-s + 2i·37-s − 12.7i·41-s + 7·49-s + 12.7·53-s + 10i·61-s − 16.9i·65-s − 16·73-s + 17.9i·85-s + 4.24i·89-s − 8·97-s + ⋯
L(s)  = 1  + 1.89·5-s − 1.10i·13-s + 1.02i·17-s + 2.59·25-s + 0.787·29-s + 0.328i·37-s − 1.98i·41-s + 49-s + 1.74·53-s + 1.28i·61-s − 2.10i·65-s − 1.87·73-s + 1.95i·85-s + 0.449i·89-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.678919246\)
\(L(\frac12)\) \(\approx\) \(2.678919246\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.24T + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 12.7iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−8.921914078180615208440532740573, −8.489548004770729799127592986731, −7.32662916165193858168639391288, −6.51583295633916456105894335849, −5.65087904591966964672412523318, −5.44696298790095134222291905347, −4.17354305510415569687978755613, −2.93554996838928646773357707858, −2.15102708436177572238775636195, −1.09438050715661994378433498138, 1.20716232499626113037519851034, 2.19232052052139528769633708207, 2.93329519775264030072175111275, 4.39343322099944335165981441256, 5.14541796010183028049516563376, 5.94281318274246497042621590415, 6.60854991362190984889435734538, 7.25251082852595622022636767993, 8.525695426506451296908935644634, 9.164775598741539129520216986884

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