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Properties

Label 2-39e2-13.12-c1-0-27
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  − i·2-s + 4-s − 2i·5-s + 4i·7-s − 3i·8-s − 2·10-s + 4i·11-s + 4·14-s − 16-s + 2·17-s − 2i·20-s + 4·22-s + 25-s + 4i·28-s + 10·29-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s − 0.894i·5-s + 1.51i·7-s − 1.06i·8-s − 0.632·10-s + 1.20i·11-s + 1.06·14-s − 0.250·16-s + 0.485·17-s − 0.447i·20-s + 0.852·22-s + 0.200·25-s + 0.755i·28-s + 1.85·29-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

\(L(1)\) \(\approx\) \(2.118015988\)
\(L(\frac12)\) \(\approx\) \(2.118015988\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 2T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.384373766189590463031197894900, −8.835438824409376606960205624597, −7.918147403743473727379022895859, −6.93849291761639245410735055019, −6.05535666758222217048917172906, −5.15244582722234115187458525947, −4.35761070343197201943379184790, −2.99835845276749615904355331952, −2.24083217726722598682601865848, −1.21592559322955815662451966162, 0.986715544991487796685379614942, 2.66683722585971036089659170777, 3.44424116905537115240067014073, 4.55083173295707621349601812299, 5.75401838951053768388123523620, 6.48650284417218830676544772663, 7.02005853911333667455607411882, 7.81781235133689436226841505768, 8.334248954155828532794708635046, 9.632329551395308856333565239859

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