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Properties

Label 2-72450-1.1-c1-0-123
Degree $2$
Conductor $72450$
Sign $-1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 + 7-s + 8-s + 2·11-s + 4·13-s + 14-s + 16-s − 6·17-s + 2·22-s + 23-s + 4·26-s + 28-s + 2·29-s + 4·31-s + 32-s − 6·34-s − 6·41-s − 6·43-s + 2·44-s + 46-s + 49-s + 4·52-s − 12·53-s + 56-s + 2·58-s + 10·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.426·22-s + 0.208·23-s + 0.784·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.937·41-s − 0.914·43-s + 0.301·44-s + 0.147·46-s + 1/7·49-s + 0.554·52-s − 1.64·53-s + 0.133·56-s + 0.262·58-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
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

−14.38507628924047, −13.73095106103285, −13.38477378097908, −13.05404185701825, −12.35884936403942, −11.81434797305317, −11.34172366876242, −11.09790183954324, −10.42706236705943, −9.938588185417753, −9.183720396729209, −8.668410885383558, −8.312456816678100, −7.667585861917708, −6.864233422374364, −6.507647230415568, −6.197180125396614, −5.271960879364412, −4.951045252842184, −4.103662456441137, −3.949853979646986, −3.067410078262034, −2.508371493489318, −1.640419769237324, −1.213404672634293, 0, 1.213404672634293, 1.640419769237324, 2.508371493489318, 3.067410078262034, 3.949853979646986, 4.103662456441137, 4.951045252842184, 5.271960879364412, 6.197180125396614, 6.507647230415568, 6.864233422374364, 7.667585861917708, 8.312456816678100, 8.668410885383558, 9.183720396729209, 9.938588185417753, 10.42706236705943, 11.09790183954324, 11.34172366876242, 11.81434797305317, 12.35884936403942, 13.05404185701825, 13.38477378097908, 13.73095106103285, 14.38507628924047

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