↑

Properties

Label 2-3549-3549.629-c0-0-1
Degree $2$
Conductor $3549$
Sign $-0.613 + 0.790i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.239 − 0.970i)3-s + (0.822 − 0.568i)4-s + (0.120 − 0.992i)7-s + (−0.885 − 0.464i)9-s + (−0.354 − 0.935i)12-s + i·13-s + (0.354 − 0.935i)16-s + (−0.420 − 0.420i)19-s + (−0.935 − 0.354i)21-s + (−0.663 − 0.748i)25-s + (−0.663 + 0.748i)27-s + (−0.464 − 0.885i)28-s + (−0.0217 − 0.359i)31-s + (−0.992 + 0.120i)36-s + (−0.0744 − 1.23i)37-s + ⋯
L(s)  = 1  + (0.239 − 0.970i)3-s + (0.822 − 0.568i)4-s + (0.120 − 0.992i)7-s + (−0.885 − 0.464i)9-s + (−0.354 − 0.935i)12-s + i·13-s + (0.354 − 0.935i)16-s + (−0.420 − 0.420i)19-s + (−0.935 − 0.354i)21-s + (−0.663 − 0.748i)25-s + (−0.663 + 0.748i)27-s + (−0.464 − 0.885i)28-s + (−0.0217 − 0.359i)31-s + (−0.992 + 0.120i)36-s + (−0.0744 − 1.23i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.613 + 0.790i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.613 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.557886447\)
\(L(\frac12)\) \(\approx\) \(1.557886447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.239 + 0.970i)T \)
7 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 - iT \)
good2 \( 1 + (-0.822 + 0.568i)T^{2} \)
5 \( 1 + (0.663 + 0.748i)T^{2} \)
11 \( 1 + (-0.822 - 0.568i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (0.420 + 0.420i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.568 - 0.822i)T^{2} \)
31 \( 1 + (0.0217 + 0.359i)T + (-0.992 + 0.120i)T^{2} \)
37 \( 1 + (0.0744 + 1.23i)T + (-0.992 + 0.120i)T^{2} \)
41 \( 1 + (0.464 - 0.885i)T^{2} \)
43 \( 1 + (-1.31 - 1.48i)T + (-0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.935 - 0.354i)T^{2} \)
53 \( 1 + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (-0.663 - 0.748i)T^{2} \)
61 \( 1 + (-1.12 - 0.136i)T + (0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.0217 + 0.118i)T + (-0.935 + 0.354i)T^{2} \)
71 \( 1 + (-0.464 + 0.885i)T^{2} \)
73 \( 1 + (-0.244 + 0.783i)T + (-0.822 - 0.568i)T^{2} \)
79 \( 1 + (0.271 - 0.393i)T + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + (-0.464 - 0.885i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.28 - 0.580i)T + (0.663 - 0.748i)T^{2} \)
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

−8.268750251287179228985113538436, −7.53659931155841534264493193975, −7.03862714496533735071482752640, −6.38612162169350288047421023244, −5.80832753409452465950409028074, −4.63734245773377408383085773700, −3.74561357470116695676058414957, −2.55600582942563242309749738647, −1.88222445614007283763313036205, −0.841930588672628310663563241272, 1.91610505653359793522491504573, 2.77861280389828733654607008687, 3.40893257825016992706201311684, 4.28006881664945999839718341667, 5.44069970272495910360024269667, 5.75727357757083181960011806169, 6.78347427401929151211391578331, 7.76113586452053505451103881305, 8.320698707860465939839251185016, 8.864031877464220520010781036131

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