Properties

Label 2-1078-1.1-c1-0-22
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 $0$

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 − 2·13-s + 4·15-s + 16-s + 18-s + 2·19-s + 2·20-s + 22-s + 2·24-s − 25-s − 2·26-s − 4·27-s + 6·29-s + 4·30-s − 4·31-s + 32-s + 2·33-s + 36-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 − 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.447·20-s + 0.213·22-s + 0.408·24-s − 1/5·25-s − 0.392·26-s − 0.769·27-s + 1.11·29-s + 0.730·30-s − 0.718·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-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: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.070208681\)
\(L(\frac12)\) \(\approx\) \(4.070208681\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−9.713147363393315144435048431072, −9.175423682574950979075679316769, −8.203128607620911587990089331160, −7.40610219486607321068429060678, −6.42499188335472548897760109457, −5.57325085629946089824969447640, −4.59150370309827712818722754536, −3.45806836703144407521124138606, −2.63884179417720314526766713244, −1.71359486469459319942838085878, 1.71359486469459319942838085878, 2.63884179417720314526766713244, 3.45806836703144407521124138606, 4.59150370309827712818722754536, 5.57325085629946089824969447640, 6.42499188335472548897760109457, 7.40610219486607321068429060678, 8.203128607620911587990089331160, 9.175423682574950979075679316769, 9.713147363393315144435048431072

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