↑

Properties

Label 2-1078-7.4-c1-0-3
Degree $2$
Conductor $1078$
Sign $-0.605 - 0.795i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.41 − 2.44i)5-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.41 − 2.44i)10-s + (−0.5 + 0.866i)11-s − 5.65·13-s + (−0.5 − 0.866i)16-s + (1.41 − 2.44i)17-s + (−1.5 + 2.59i)18-s + (4.24 + 7.34i)19-s + 2.82·20-s − 0.999·22-s + (4 + 6.92i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.632 − 1.09i)5-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.447 − 0.774i)10-s + (−0.150 + 0.261i)11-s − 1.56·13-s + (−0.125 − 0.216i)16-s + (0.342 − 0.594i)17-s + (−0.353 + 0.612i)18-s + (0.973 + 1.68i)19-s + 0.632·20-s − 0.213·22-s + (0.834 + 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.163836721\)
\(L(\frac12)\) \(\approx\) \(1.163836721\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 7.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4.24 - 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.82 - 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (4.24 - 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + (-5.65 - 9.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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

−9.829830415618208863815785975058, −9.413367222499596184376416519181, −8.228592552766883094168777485754, −7.52056555745825945939714066809, −7.25431588565736251856680769295, −5.51635620883250021461696594642, −5.11379231616085763030492813835, −4.32397177036322888782790060971, −3.17528496827310334658313984916, −1.52711398307412433780369009222, 0.47091112443030590078212435359, 2.41418343712792240296525227159, 3.17192163934378238436428927005, 4.12232062349114558505441557990, 5.06729461168235260334755207169, 6.25848922324922639371056781300, 7.16120151184465327456370574991, 7.63057660440348854854983408918, 9.139171584939684717806923604655, 9.616883414583547896249114625400

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