↑

Properties

Label 2-1001-7.2-c1-0-68
Degree $2$
Conductor $1001$
Sign $-0.998 - 0.0538i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
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  + (0.608 − 1.05i)2-s + (−1.54 − 2.67i)3-s + (0.259 + 0.448i)4-s + (1.72 − 2.98i)5-s − 3.75·6-s + (2.12 + 1.58i)7-s + 3.06·8-s + (−3.25 + 5.63i)9-s + (−2.09 − 3.63i)10-s + (−0.5 − 0.866i)11-s + (0.798 − 1.38i)12-s − 13-s + (2.95 − 1.27i)14-s − 10.6·15-s + (1.34 − 2.33i)16-s + (−3.43 − 5.94i)17-s + ⋯
L(s)  = 1  + (0.430 − 0.745i)2-s + (−0.890 − 1.54i)3-s + (0.129 + 0.224i)4-s + (0.770 − 1.33i)5-s − 1.53·6-s + (0.801 + 0.597i)7-s + 1.08·8-s + (−1.08 + 1.87i)9-s + (−0.662 − 1.14i)10-s + (−0.150 − 0.261i)11-s + (0.230 − 0.399i)12-s − 0.277·13-s + (0.790 − 0.340i)14-s − 2.74·15-s + (0.336 − 0.583i)16-s + (−0.832 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.998 - 0.0538i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ -0.998 - 0.0538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850545827\)
\(L(\frac12)\) \(\approx\) \(1.850545827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.12 - 1.58i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.608 + 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.54 + 2.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (3.43 + 5.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.17 - 2.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.18 + 5.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 + (3.16 + 5.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.10 - 5.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 0.341T + 43T^{2} \)
47 \( 1 + (0.376 - 0.651i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.95 - 5.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.05 + 1.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.63 + 9.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.21 + 2.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0325T + 71T^{2} \)
73 \( 1 + (-6.53 - 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.31 + 2.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.0974T + 83T^{2} \)
89 \( 1 + (2.82 - 4.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.32T + 97T^{2} \)
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

−9.513765259753282916106253249035, −8.588063093543677335421362233999, −7.82447852325187665909974650555, −7.02588132765462506640050081980, −5.91030559015240306077555138678, −5.19360766952942436341141034387, −4.54180102249449807839551231692, −2.45659990731171917788730828870, −1.93045223347959130810353636841, −0.820465517912102587541704906819, 1.96141843496533247944093194707, 3.65308485817303935664802875697, 4.44398133201168192883143306516, 5.40267925693177033791455435311, 5.86850813935330441614509287053, 6.81925908180676384589441103183, 7.44323467231373950638714432830, 8.983894679328811740275806807359, 9.903280422582809013640001970669, 10.56954394960092832799233065568

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