↑

Properties

Label 2-1078-77.76-c1-0-14
Degree $2$
Conductor $1078$
Sign $-0.0922 - 0.995i$
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  + i·2-s − 0.765i·3-s − 4-s + 3.89i·5-s + 0.765·6-s − i·8-s + 2.41·9-s − 3.89·10-s + (0.214 − 3.30i)11-s + 0.765i·12-s + 1.81·13-s + 2.98·15-s + 16-s + 6.07·17-s + 2.41i·18-s + 6.82·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.441i·3-s − 0.5·4-s + 1.74i·5-s + 0.312·6-s − 0.353i·8-s + 0.804·9-s − 1.23·10-s + (0.0647 − 0.997i)11-s + 0.220i·12-s + 0.504·13-s + 0.770·15-s + 0.250·16-s + 1.47·17-s + 0.569i·18-s + 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0922 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0922 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.0922 - 0.995i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.0922 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.714797457\)
\(L(\frac12)\) \(\approx\) \(1.714797457\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (-0.214 + 3.30i)T \)
good3 \( 1 + 0.765iT - 3T^{2} \)
5 \( 1 - 3.89iT - 5T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 - 9.36iT - 29T^{2} \)
31 \( 1 - 7.88iT - 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 + 3.25iT - 43T^{2} \)
47 \( 1 - 0.203iT - 47T^{2} \)
53 \( 1 - 4.98T + 53T^{2} \)
59 \( 1 - 5.19iT - 59T^{2} \)
61 \( 1 + 8.98T + 61T^{2} \)
67 \( 1 + 8.03T + 67T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + 0.0557T + 73T^{2} \)
79 \( 1 + 8.48iT - 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + 1.68iT - 89T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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

−10.32984516940894801671627904571, −9.233770434931576004887796371879, −8.119269284582195046954829289985, −7.37957986122028217312612753627, −6.89735805727399176337864844776, −6.06840325963211694026551368786, −5.27514744038724236888352944556, −3.57081991730719960720414453023, −3.18706112267046851794420388837, −1.37833599692662527291451119480, 0.929652384232676273942279992610, 1.86690043279335628738825835517, 3.62704593878762030784461898622, 4.34128172536921731002509410602, 5.08278507480582089450924033607, 5.87196182849699053680167676409, 7.56338957780341744414639964887, 8.068972446789505176757803320769, 9.188319913684649931003635972255, 9.843917447297248135691045107100

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