Properties

Label 2-1078-1.1-c1-0-28
Degree $2$
Conductor $1078$
Sign $-1$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 11-s + 12-s − 13-s + 16-s − 6·17-s + 2·18-s + 2·19-s + 22-s − 6·23-s − 24-s − 5·25-s + 26-s − 5·27-s + 9·29-s − 4·31-s − 32-s − 33-s + 6·34-s − 2·36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 0.458·19-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 25-s + 0.196·26-s − 0.962·27-s + 1.67·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s + 1.02·34-s − 1/3·36-s + 0.328·37-s − 0.324·38-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: yes
Arithmetic: yes
Character: $\chi_{1078} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.515724472288571684089286227040, −8.364564560362548777090135115264, −8.265521675023629977633370602556, −7.09577818778194036472308733013, −6.26509669227661442964747384857, −5.23231327938266363285847800045, −3.97281851054077595519569236837, −2.80907868912292619472481812542, −1.95031804062996461477464635472, 0, 1.95031804062996461477464635472, 2.80907868912292619472481812542, 3.97281851054077595519569236837, 5.23231327938266363285847800045, 6.26509669227661442964747384857, 7.09577818778194036472308733013, 8.265521675023629977633370602556, 8.364564560362548777090135115264, 9.515724472288571684089286227040

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