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Properties

Label 2-12e3-8.5-c1-0-16
Degree $2$
Conductor $1728$
Sign $0.707 - 0.707i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  + 3i·5-s + 3·7-s + 3i·11-s − 6i·13-s + 6·17-s − 2i·19-s + 6·23-s − 4·25-s − 6i·29-s + 3·31-s + 9i·35-s + 6i·37-s + 6·41-s + 8i·43-s − 12·47-s + ⋯
L(s)  = 1  + 1.34i·5-s + 1.13·7-s + 0.904i·11-s − 1.66i·13-s + 1.45·17-s − 0.458i·19-s + 1.25·23-s − 0.800·25-s − 1.11i·29-s + 0.538·31-s + 1.52i·35-s + 0.986i·37-s + 0.937·41-s + 1.21i·43-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.124225977\)
\(L(\frac12)\) \(\approx\) \(2.124225977\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−9.778167774539802117147448433113, −8.335705832750782116259770359924, −7.77572782486253416961356003743, −7.21341749490191112740640486581, −6.24333476053035838603584058915, −5.31701716934340813953930178113, −4.56051684217677194176409927739, −3.21165482118584790844721391666, −2.65157290845513525608423542709, −1.21409406908452432742353363030, 1.02738354571215150142269014578, 1.77025280910176116520663795636, 3.37911633933529852951966615371, 4.42236060216706181384132011387, 5.07860047042322202349512194047, 5.73870367187247599693109243408, 6.91242470636671259427802672516, 7.84129509268527927250600321243, 8.554463512971505208332575214289, 9.006012659423267900666190988806

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