↑

Properties

Label 2-1045-209.208-c1-0-35
Degree $2$
Conductor $1045$
Sign $-0.440 + 0.897i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  − 2.08·2-s − 2.80i·3-s + 2.35·4-s − 5-s + 5.85i·6-s + 0.950i·7-s − 0.737·8-s − 4.88·9-s + 2.08·10-s + (0.932 + 3.18i)11-s − 6.60i·12-s + 3.48·13-s − 1.98i·14-s + 2.80i·15-s − 3.16·16-s + 1.26i·17-s + ⋯
L(s)  = 1  − 1.47·2-s − 1.62i·3-s + 1.17·4-s − 0.447·5-s + 2.39i·6-s + 0.359i·7-s − 0.260·8-s − 1.62·9-s + 0.659·10-s + (0.281 + 0.959i)11-s − 1.90i·12-s + 0.967·13-s − 0.529i·14-s + 0.725i·15-s − 0.791·16-s + 0.306i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.440 + 0.897i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6287983273\)
\(L(\frac12)\) \(\approx\) \(0.6287983273\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + (-0.932 - 3.18i)T \)
19 \( 1 + (3.21 + 2.94i)T \)
good2 \( 1 + 2.08T + 2T^{2} \)
3 \( 1 + 2.80iT - 3T^{2} \)
7 \( 1 - 0.950iT - 7T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 - 1.26iT - 17T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + 6.79iT - 31T^{2} \)
37 \( 1 + 3.59iT - 37T^{2} \)
41 \( 1 + 3.33T + 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 - 9.25T + 47T^{2} \)
53 \( 1 + 3.95iT - 53T^{2} \)
59 \( 1 + 3.82iT - 59T^{2} \)
61 \( 1 + 3.98iT - 61T^{2} \)
67 \( 1 - 7.67iT - 67T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + 6.81iT - 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 + 4.45iT - 89T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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

−9.335279957905743622938866054190, −8.602619871496865528390152999336, −8.148314406334663578206907179777, −7.12302829477455861768033759197, −6.90490841543178307781442135877, −5.84731873175009123297020681920, −4.30219707379311768033262121217, −2.57585389284589498633780668207, −1.68117823442086949037944861562, −0.61013315124831757070095664362, 1.03553380467506533226540914572, 3.03274786679255509277920264591, 3.93085787856992050281764752234, 4.80571744805432433101908988277, 6.03210921525133362150275704507, 7.06544139011474334271434269744, 8.221604933121803255588618046813, 8.759310541711859914328180669382, 9.200584207482840942975302359997, 10.33682768224057333052318435701

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