Properties

Label 2-348726-1.1-c1-0-27
Degree $2$
Conductor $348726$
Sign $-1$
Analytic cond. $2784.59$
Root an. cond. $52.7692$
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 + 7-s − 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s − 6·17-s − 18-s + 21-s − 23-s − 24-s − 5·25-s + 2·26-s + 27-s + 28-s + 6·29-s − 8·31-s − 32-s + 6·34-s + 36-s + 6·37-s − 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348726\)    =    \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2784.59\)
Root analytic conductor: \(52.7692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{348726} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ -1)\)

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

−12.95305576986278, −12.17901261274839, −11.73361531635488, −11.47858837345999, −10.90272926150349, −10.37795149643493, −10.03603040806168, −9.565596664330943, −9.079349255442770, −8.641807332719914, −8.304050263602305, −7.808713699913052, −7.346219487691525, −6.844845595705444, −6.490775000592652, −5.883995291277703, −5.204407987270334, −4.774160669385924, −4.203695318886310, −3.644245646777700, −3.102459080762656, −2.364984631849064, −2.044909723705088, −1.581402209804436, −0.6750727228408207, 0, 0.6750727228408207, 1.581402209804436, 2.044909723705088, 2.364984631849064, 3.102459080762656, 3.644245646777700, 4.203695318886310, 4.774160669385924, 5.204407987270334, 5.883995291277703, 6.490775000592652, 6.844845595705444, 7.346219487691525, 7.808713699913052, 8.304050263602305, 8.641807332719914, 9.079349255442770, 9.565596664330943, 10.03603040806168, 10.37795149643493, 10.90272926150349, 11.47858837345999, 11.73361531635488, 12.17901261274839, 12.95305576986278

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