↑

Properties

Label 2-1045-5.4-c1-0-10
Degree $2$
Conductor $1045$
Sign $-0.261 - 0.965i$
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  + 2.14i·2-s − 2.93i·3-s − 2.60·4-s + (−0.584 − 2.15i)5-s + 6.30·6-s + 3.64i·7-s − 1.30i·8-s − 5.62·9-s + (4.63 − 1.25i)10-s + 11-s + 7.66i·12-s + 3.27i·13-s − 7.81·14-s + (−6.33 + 1.71i)15-s − 2.40·16-s + 1.25i·17-s + ⋯
L(s)  = 1  + 1.51i·2-s − 1.69i·3-s − 1.30·4-s + (−0.261 − 0.965i)5-s + 2.57·6-s + 1.37i·7-s − 0.462i·8-s − 1.87·9-s + (1.46 − 0.396i)10-s + 0.301·11-s + 2.21i·12-s + 0.907i·13-s − 2.08·14-s + (−1.63 + 0.443i)15-s − 0.602·16-s + 0.303i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.261 - 0.965i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151641174\)
\(L(\frac12)\) \(\approx\) \(1.151641174\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.584 + 2.15i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 2.14iT - 2T^{2} \)
3 \( 1 + 2.93iT - 3T^{2} \)
7 \( 1 - 3.64iT - 7T^{2} \)
13 \( 1 - 3.27iT - 13T^{2} \)
17 \( 1 - 1.25iT - 17T^{2} \)
23 \( 1 - 5.03iT - 23T^{2} \)
29 \( 1 - 3.55T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 - 7.02iT - 37T^{2} \)
41 \( 1 - 8.43T + 41T^{2} \)
43 \( 1 - 8.40iT - 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 2.22iT - 53T^{2} \)
59 \( 1 + 6.36T + 59T^{2} \)
61 \( 1 - 2.21T + 61T^{2} \)
67 \( 1 - 0.512iT - 67T^{2} \)
71 \( 1 + 6.75T + 71T^{2} \)
73 \( 1 - 4.03iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 0.822iT - 97T^{2} \)
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

−9.530137847755875453032515159983, −8.765138355282774440331024689657, −8.293532810821259719115016892529, −7.69481964588604734755446151338, −6.68187113510355642535092916689, −6.18052966281192331683539304536, −5.45556053587530594711025586853, −4.48300391911043168764634658131, −2.53679291449678144315592024692, −1.35331442264998124996838076462, 0.56118305671246496395157133647, 2.68257355603192255390229411725, 3.30836740703679206218872389507, 4.22177349416330258350832057388, 4.54017749234625631098282541752, 6.10218422470873394484562882214, 7.25050580369353900469552134087, 8.386227066420812745998007350785, 9.422560444193234346587696530153, 10.13886699092463415888984614077

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