Properties

Label 2-72450-1.1-c1-0-52
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 + 4·11-s − 4·13-s + 14-s + 16-s − 8·17-s − 2·19-s − 4·22-s + 23-s + 4·26-s − 28-s − 2·29-s − 6·31-s − 32-s + 8·34-s + 10·37-s + 2·38-s − 6·41-s + 8·43-s + 4·44-s − 46-s + 6·47-s + 49-s − 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s + 0.784·26-s − 0.188·28-s − 0.371·29-s − 1.07·31-s − 0.176·32-s + 1.37·34-s + 1.64·37-s + 0.324·38-s − 0.937·41-s + 1.21·43-s + 0.603·44-s − 0.147·46-s + 0.875·47-s + 1/7·49-s − 0.554·52-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: $\chi_{72450} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−14.54235601674582, −13.85627005532077, −13.30619472788067, −12.72057465288680, −12.40819144982117, −11.72251602009976, −11.18938482692756, −10.99411447704185, −10.20742906674937, −9.708866385417657, −9.209862293243090, −8.918521628026152, −8.404700498170657, −7.584939472981258, −7.138818866316197, −6.721583481339550, −6.214407161066574, −5.620042377624317, −4.808640769641567, −4.177002167302508, −3.801164457178515, −2.777727412500854, −2.317062534028502, −1.718530442877149, −0.7575889121469554, 0, 0.7575889121469554, 1.718530442877149, 2.317062534028502, 2.777727412500854, 3.801164457178515, 4.177002167302508, 4.808640769641567, 5.620042377624317, 6.214407161066574, 6.721583481339550, 7.138818866316197, 7.584939472981258, 8.404700498170657, 8.918521628026152, 9.209862293243090, 9.708866385417657, 10.20742906674937, 10.99411447704185, 11.18938482692756, 11.72251602009976, 12.40819144982117, 12.72057465288680, 13.30619472788067, 13.85627005532077, 14.54235601674582

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