↑

Properties

Label 2-3234-77.76-c1-0-28
Degree $2$
Conductor $3234$
Sign $-0.484 - 0.874i$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − i·3-s − 4-s + 3.26i·5-s + 6-s − i·8-s − 9-s − 3.26·10-s + (3.11 + 1.14i)11-s + i·12-s − 5.12·13-s + 3.26·15-s + 16-s + 5.32·17-s − i·18-s + 4.27·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.46i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.03·10-s + (0.939 + 0.343i)11-s + 0.288i·12-s − 1.42·13-s + 0.843·15-s + 0.250·16-s + 1.29·17-s − 0.235i·18-s + 0.980·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3234} (2155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ -0.484 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.707326678\)
\(L(\frac12)\) \(\approx\) \(1.707326678\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (-3.11 - 1.14i)T \)
good5 \( 1 - 3.26iT - 5T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 4.77iT - 29T^{2} \)
31 \( 1 + 6.46iT - 31T^{2} \)
37 \( 1 - 2.13T + 37T^{2} \)
41 \( 1 - 0.949T + 41T^{2} \)
43 \( 1 + 6.20iT - 43T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 13.4iT - 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 - 6.19T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 9.16iT - 79T^{2} \)
83 \( 1 - 0.835T + 83T^{2} \)
89 \( 1 + 6.22iT - 89T^{2} \)
97 \( 1 - 0.624iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.868904342977866887801713070629, −7.66091196939182486429351702570, −7.29017467432220431094182539861, −6.96753726124425460194485048375, −6.00922528447052070127720478043, −5.38483147474023481425802321481, −4.28640213844915408500800257406, −3.21821675916882805539552535612, −2.58204535349137364015242307240, −1.15676097543878880381617712638, 0.62873229384622904548981506846, 1.48895179608802446226984828108, 2.85495053839182037360593140776, 3.67851599552987330358153341250, 4.59347253373647995541816470514, 5.12570452809992234342958882529, 5.70887469272789457587000442775, 7.03619526548996152609996088676, 7.921858924469505152344268529049, 8.683155608229357641872529587797

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