↑

Properties

Label 2-1078-77.76-c1-0-7
Degree $2$
Conductor $1078$
Sign $-0.533 + 0.845i$
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 + 1.84i·3-s − 4-s + 3.23i·5-s − 1.84·6-s − i·8-s − 0.414·9-s − 3.23·10-s + (3.28 + 0.468i)11-s − 1.84i·12-s − 6.10·13-s − 5.98·15-s + 16-s − 8.13·17-s − 0.414i·18-s + 6.61·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.06i·3-s − 0.5·4-s + 1.44i·5-s − 0.754·6-s − 0.353i·8-s − 0.138·9-s − 1.02·10-s + (0.989 + 0.141i)11-s − 0.533i·12-s − 1.69·13-s − 1.54·15-s + 0.250·16-s − 1.97·17-s − 0.0976i·18-s + 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.533 + 0.845i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067513706\)
\(L(\frac12)\) \(\approx\) \(1.067513706\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 - 0.468i)T \)
good3 \( 1 - 1.84iT - 3T^{2} \)
5 \( 1 - 3.23iT - 5T^{2} \)
13 \( 1 + 6.10T + 13T^{2} \)
17 \( 1 + 8.13T + 17T^{2} \)
19 \( 1 - 6.61T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 - 1.32iT - 29T^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 - 5.28T + 37T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 + 3.73iT - 43T^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 + 3.98T + 53T^{2} \)
59 \( 1 + 9.03iT - 59T^{2} \)
61 \( 1 - 8.55T + 61T^{2} \)
67 \( 1 - 2.35T + 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 5.60T + 83T^{2} \)
89 \( 1 + 4.66iT - 89T^{2} \)
97 \( 1 - 3.82iT - 97T^{2} \)
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

−10.12956947274664834432783855803, −9.680590891470037549172536832995, −9.024389386304608141017335951427, −7.69315438041585846648997431152, −6.98368200571582544607402845978, −6.44735323650968359595975643517, −5.19048633866360451089200351506, −4.37286802456165746174645776610, −3.53532617017366248311543314640, −2.35612711403107512679153491749, 0.47575819445438652828113327248, 1.57496718668128786031841974846, 2.47278051039369015064336065290, 4.18167207549052124229138580386, 4.74123109523784716445147550240, 5.86837524172682359741002793392, 6.94810310870929442501939023928, 7.75780598861123138180131619020, 8.591584522957761186257425833190, 9.444411615957849902174860255457

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