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Properties

Label 2-39e2-13.12-c1-0-39
Degree $2$
Conductor $1521$
Sign $0.832 + 0.554i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
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  + 2·4-s − i·7-s + 4·16-s − 8i·19-s + 5·25-s − 2i·28-s + 7i·31-s − 10i·37-s + 13·43-s + 6·49-s − 13·61-s + 8·64-s − 11i·67-s + 17i·73-s − 16i·76-s + ⋯
L(s)  = 1  + 4-s − 0.377i·7-s + 16-s − 1.83i·19-s + 25-s − 0.377i·28-s + 1.25i·31-s − 1.64i·37-s + 1.98·43-s + 0.857·49-s − 1.66·61-s + 64-s − 1.34i·67-s + 1.98i·73-s − 1.83i·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.213805537\)
\(L(\frac12)\) \(\approx\) \(2.213805537\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7iT - 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 17iT - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−9.295072359919803899908115580499, −8.684226119232464045011621180990, −7.45384282831554294061437511132, −7.12396746697999620413722732723, −6.26536375076026176873066859954, −5.31671394149868535975107086300, −4.32970111751942568956851864316, −3.13743671552165591930356222847, −2.33861750099995445869561081693, −0.951656736640315526814882379559, 1.35067390714073220600406077605, 2.44074225222256654961317362096, 3.36033846437619319020478767115, 4.47155518430251494812864844911, 5.77147114024431785919914255069, 6.11660382041295657273521805962, 7.19538398166026916977415403049, 7.85110923973955744832581489947, 8.641084654015650009378405767515, 9.659082610768094433200294304598

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