Properties

Label 2-348726-1.1-c1-0-23
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 − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s + 14-s + 2·15-s + 16-s + 6·17-s − 18-s − 2·20-s + 21-s − 4·22-s + 23-s + 24-s − 25-s + 2·26-s − 27-s − 28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.59618232116738, −12.22307665989001, −11.85255670757569, −11.42005078507404, −10.96241740368680, −10.64112595280067, −9.808825887949783, −9.713085606551284, −9.302031965539909, −8.590921724538121, −8.237825690055798, −7.633173313708557, −7.212130189435335, −7.030622032257700, −6.188853632949923, −6.009142594848970, −5.351250052520131, −4.764934817534973, −4.168998844837145, −3.688009427152086, −3.246693677582704, −2.643576308900133, −1.758663401815196, −1.276171012874049, −0.6446124021159075, 0, 0.6446124021159075, 1.276171012874049, 1.758663401815196, 2.643576308900133, 3.246693677582704, 3.688009427152086, 4.168998844837145, 4.764934817534973, 5.351250052520131, 6.009142594848970, 6.188853632949923, 7.030622032257700, 7.212130189435335, 7.633173313708557, 8.237825690055798, 8.590921724538121, 9.302031965539909, 9.713085606551284, 9.808825887949783, 10.64112595280067, 10.96241740368680, 11.42005078507404, 11.85255670757569, 12.22307665989001, 12.59618232116738

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