Properties

Label 2-2576-161.160-c1-0-48
Degree $2$
Conductor $2576$
Sign $0.625 - 0.780i$
Analytic cond. $20.5694$
Root an. cond. $4.53535$
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  + 1.41i·3-s + 2.64·7-s + 0.999·9-s + 3.74i·11-s − 4.24i·13-s + 5.29·19-s + 3.74i·21-s + (3 − 3.74i)23-s − 5·25-s + 5.65i·27-s + 6·29-s + 8.48i·31-s − 5.29·33-s − 11.2i·37-s + 6·39-s + ⋯
L(s)  = 1  + 0.816i·3-s + 0.999·7-s + 0.333·9-s + 1.12i·11-s − 1.17i·13-s + 1.21·19-s + 0.816i·21-s + (0.625 − 0.780i)23-s − 25-s + 1.08i·27-s + 1.11·29-s + 1.52i·31-s − 0.921·33-s − 1.84i·37-s + 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(20.5694\)
Root analytic conductor: \(4.53535\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :1/2),\ 0.625 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.311116202\)
\(L(\frac12)\) \(\approx\) \(2.311116202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 2.64T \)
23 \( 1 + (-3 + 3.74i)T \)
good3 \( 1 - 1.41iT - 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 3.74iT - 53T^{2} \)
59 \( 1 + 9.89iT - 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−9.023417951417864507615499745295, −8.284775894771633193920278377110, −7.45043269398359145531812674296, −6.93127238582756090319041990570, −5.50797313920747180941834411383, −5.05916529625712966512513547019, −4.33434567855740257558573644448, −3.45080002586962300937467625592, −2.30520733186659616881336764595, −1.10773179750183835997731644017, 0.996749849920824345862613272028, 1.73186654180911346047782604561, 2.87721432859700053196714074370, 4.03176660498491586558298498061, 4.83786593985071988520371110232, 5.80832192920134348607548072825, 6.50760327051521737751408480948, 7.41038935625392119995422296561, 7.87355378228929608250183733803, 8.626501790763395071084342045740

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