↑

Properties

Label 2-1045-5.4-c1-0-87
Degree $2$
Conductor $1045$
Sign $-0.694 - 0.719i$
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  − 1.06i·2-s − 2.29i·3-s + 0.872·4-s + (−1.55 − 1.60i)5-s − 2.43·6-s − 3.97i·7-s − 3.05i·8-s − 2.25·9-s + (−1.70 + 1.64i)10-s − 11-s − 2.00i·12-s + 4.18i·13-s − 4.21·14-s + (−3.68 + 3.55i)15-s − 1.49·16-s − 2.67i·17-s + ⋯
L(s)  = 1  − 0.750i·2-s − 1.32i·3-s + 0.436·4-s + (−0.694 − 0.719i)5-s − 0.993·6-s − 1.50i·7-s − 1.07i·8-s − 0.752·9-s + (−0.540 + 0.521i)10-s − 0.301·11-s − 0.577i·12-s + 1.15i·13-s − 1.12·14-s + (−0.952 + 0.919i)15-s − 0.373·16-s − 0.648i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.694 - 0.719i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570073902\)
\(L(\frac12)\) \(\approx\) \(1.570073902\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.55 + 1.60i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 1.06iT - 2T^{2} \)
3 \( 1 + 2.29iT - 3T^{2} \)
7 \( 1 + 3.97iT - 7T^{2} \)
13 \( 1 - 4.18iT - 13T^{2} \)
17 \( 1 + 2.67iT - 17T^{2} \)
23 \( 1 - 1.19iT - 23T^{2} \)
29 \( 1 - 7.35T + 29T^{2} \)
31 \( 1 - 8.19T + 31T^{2} \)
37 \( 1 - 1.43iT - 37T^{2} \)
41 \( 1 - 1.11T + 41T^{2} \)
43 \( 1 - 2.88iT - 43T^{2} \)
47 \( 1 - 7.25iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 1.48T + 59T^{2} \)
61 \( 1 - 2.05T + 61T^{2} \)
67 \( 1 - 5.82iT - 67T^{2} \)
71 \( 1 - 2.59T + 71T^{2} \)
73 \( 1 - 5.35iT - 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 4.87iT - 97T^{2} \)
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

−9.595480216859793713286836905981, −8.320212731432198067099781145129, −7.60640645896016527408090567599, −6.96900594183000069130007413301, −6.43992722984844306707017304218, −4.73055836047201671361537427897, −3.94109611383644506836519459256, −2.72516128026025299411767851743, −1.41222244905810379569981897032, −0.75379899604291326400472811769, 2.56724226179754549741362787976, 3.16875940757089263907025734107, 4.49599081148734584970961648666, 5.42674276664866516543134027244, 6.05152653498310971748474448003, 7.03360275779076479176642699863, 8.270806835474286713714350954623, 8.389721202204356573867617224969, 9.679609826405745300403352817402, 10.52940367030831267573649357614

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