Properties

Label 2-1078-1.1-c1-0-13
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 − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 9-s + 2·10-s + 11-s − 2·12-s + 4·13-s + 4·15-s + 16-s − 18-s − 4·19-s − 2·20-s − 22-s + 4·23-s + 2·24-s − 25-s − 4·26-s + 4·27-s + 2·29-s − 4·30-s + 10·31-s − 32-s − 2·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.213·22-s + 0.834·23-s + 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.371·29-s − 0.730·30-s + 1.79·31-s − 0.176·32-s − 0.348·33-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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.512353895122736054642994261496, −8.438279870467851105214692730660, −8.059886585841006026613889387391, −6.63986065044552740769645726528, −6.45975539812673080939370676867, −5.23639126138259380433417465369, −4.24221631118951556259235272721, −3.09515527693911842740827286057, −1.30846072392587640239816399795, 0, 1.30846072392587640239816399795, 3.09515527693911842740827286057, 4.24221631118951556259235272721, 5.23639126138259380433417465369, 6.45975539812673080939370676867, 6.63986065044552740769645726528, 8.059886585841006026613889387391, 8.438279870467851105214692730660, 9.512353895122736054642994261496

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