Properties

Label 2-69678-1.1-c1-0-10
Degree $2$
Conductor $69678$
Sign $-1$
Analytic cond. $556.381$
Root an. cond. $23.5877$
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 + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s − 2·13-s + 16-s − 2·17-s − 2·20-s − 4·22-s − 25-s + 2·26-s − 8·29-s − 8·31-s − 32-s + 2·34-s + 4·37-s + 2·40-s − 10·41-s − 2·43-s + 4·44-s + 50-s − 2·52-s + 8·53-s − 8·55-s + 8·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s − 1.48·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s + 0.316·40-s − 1.56·41-s − 0.304·43-s + 0.603·44-s + 0.141·50-s − 0.277·52-s + 1.09·53-s − 1.07·55-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69678\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 79\)
Sign: $-1$
Analytic conductor: \(556.381\)
Root analytic conductor: \(23.5877\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 69678,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
79 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.69674385773044, −13.95049559845017, −13.31553040510364, −12.83154477081239, −12.19305901420533, −11.76938127627145, −11.34515763877006, −11.06807674772346, −10.25933523328283, −9.768333343241987, −9.310772687362020, −8.682771727325343, −8.456727010141663, −7.551529336368969, −7.338322007211144, −6.820749181195385, −6.204331863445010, −5.522602474901016, −4.928042575422276, −3.961838652717013, −3.862293681645247, −3.112479264406057, −2.130389252622517, −1.729509488950422, −0.7279197459888474, 0, 0.7279197459888474, 1.729509488950422, 2.130389252622517, 3.112479264406057, 3.862293681645247, 3.961838652717013, 4.928042575422276, 5.522602474901016, 6.204331863445010, 6.820749181195385, 7.338322007211144, 7.551529336368969, 8.456727010141663, 8.682771727325343, 9.310772687362020, 9.768333343241987, 10.25933523328283, 11.06807674772346, 11.34515763877006, 11.76938127627145, 12.19305901420533, 12.83154477081239, 13.31553040510364, 13.95049559845017, 14.69674385773044

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