Properties

Label 2-1078-1.1-c1-0-18
Degree $2$
Conductor $1078$
Sign $1$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3.23·3-s + 4-s − 3.23·5-s + 3.23·6-s + 8-s + 7.47·9-s − 3.23·10-s + 11-s + 3.23·12-s − 1.23·13-s − 10.4·15-s + 16-s + 6.47·17-s + 7.47·18-s + 2.76·19-s − 3.23·20-s + 22-s + 4·23-s + 3.23·24-s + 5.47·25-s − 1.23·26-s + 14.4·27-s − 4.47·29-s − 10.4·30-s − 2·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.86·3-s + 0.5·4-s − 1.44·5-s + 1.32·6-s + 0.353·8-s + 2.49·9-s − 1.02·10-s + 0.301·11-s + 0.934·12-s − 0.342·13-s − 2.70·15-s + 0.250·16-s + 1.56·17-s + 1.76·18-s + 0.634·19-s − 0.723·20-s + 0.213·22-s + 0.834·23-s + 0.660·24-s + 1.09·25-s − 0.242·26-s + 2.78·27-s − 0.830·29-s − 1.91·30-s − 0.359·31-s + 0.176·32-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.939728353\)
\(L(\frac12)\) \(\approx\) \(3.939728353\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 3.23T + 3T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 2.76T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 + 7.23T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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

−9.743913546027839964224405485591, −8.868440804206357708140504609090, −8.107011218752175823817849183982, −7.42804411366250484970207537346, −7.03000964499145678562181388468, −5.28717239674767381762556781599, −4.24663132650566531859521090082, −3.43146858905407759239798601585, −3.08791371561946288993837862507, −1.55719080927706853210503358377, 1.55719080927706853210503358377, 3.08791371561946288993837862507, 3.43146858905407759239798601585, 4.24663132650566531859521090082, 5.28717239674767381762556781599, 7.03000964499145678562181388468, 7.42804411366250484970207537346, 8.107011218752175823817849183982, 8.868440804206357708140504609090, 9.743913546027839964224405485591

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