Properties

Label 2-1078-1.1-c1-0-16
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.64·3-s + 4-s + 1.64·5-s + 2.64·6-s − 8-s + 4.00·9-s − 1.64·10-s + 11-s − 2.64·12-s − 5·13-s − 4.35·15-s + 16-s − 6·17-s − 4.00·18-s + 5.64·19-s + 1.64·20-s − 22-s + 1.64·23-s + 2.64·24-s − 2.29·25-s + 5·26-s − 2.64·27-s + 6.29·29-s + 4.35·30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.52·3-s + 0.5·4-s + 0.736·5-s + 1.08·6-s − 0.353·8-s + 1.33·9-s − 0.520·10-s + 0.301·11-s − 0.763·12-s − 1.38·13-s − 1.12·15-s + 0.250·16-s − 1.45·17-s − 0.942·18-s + 1.29·19-s + 0.368·20-s − 0.213·22-s + 0.343·23-s + 0.540·24-s − 0.458·25-s + 0.980·26-s − 0.509·27-s + 1.16·29-s + 0.794·30-s + 0.718·31-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: Trivial
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 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 3.64T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 - 1.64T + 53T^{2} \)
59 \( 1 + 4.64T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 4.35T + 71T^{2} \)
73 \( 1 + 0.354T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + 6.58T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.742799320054153365594628020594, −8.861903914239012580950561657857, −7.63567773362778055907891187915, −6.74798075308560886012109489747, −6.26152495348524356338915236257, −5.23195943579057184438072938355, −4.60836923153996931364398735448, −2.75988642619408393458819236442, −1.45225789825843922701089451377, 0, 1.45225789825843922701089451377, 2.75988642619408393458819236442, 4.60836923153996931364398735448, 5.23195943579057184438072938355, 6.26152495348524356338915236257, 6.74798075308560886012109489747, 7.63567773362778055907891187915, 8.861903914239012580950561657857, 9.742799320054153365594628020594

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