Properties

Label 2-303450-1.1-c1-0-31
Degree $2$
Conductor $303450$
Sign $1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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 + 11-s − 12-s + 5·13-s + 14-s + 16-s − 18-s − 4·19-s + 21-s − 22-s − 6·23-s + 24-s − 5·26-s − 27-s − 28-s − 4·29-s + 3·31-s − 32-s − 33-s + 36-s + 4·37-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 + 0.301·11-s − 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 0.188·28-s − 0.742·29-s + 0.538·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.246793059\)
\(L(\frac12)\) \(\approx\) \(1.246793059\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−12.74586775183861, −12.02714093819613, −11.59817649381444, −11.37823315253230, −10.80610284302741, −10.37196455511301, −9.916244611364796, −9.628036733718482, −8.920061367823458, −8.483202293387910, −8.214997158748502, −7.633488244763199, −6.937624132221360, −6.551489367482305, −6.282934262821955, −5.636449973351384, −5.376361177581416, −4.363285910279242, −4.082744744560149, −3.514426082756071, −2.936893737090987, −2.022266833937198, −1.789855552375642, −0.9282810419371737, −0.4165121670349105, 0.4165121670349105, 0.9282810419371737, 1.789855552375642, 2.022266833937198, 2.936893737090987, 3.514426082756071, 4.082744744560149, 4.363285910279242, 5.376361177581416, 5.636449973351384, 6.282934262821955, 6.551489367482305, 6.937624132221360, 7.633488244763199, 8.214997158748502, 8.483202293387910, 8.920061367823458, 9.628036733718482, 9.916244611364796, 10.37196455511301, 10.80610284302741, 11.37823315253230, 11.59817649381444, 12.02714093819613, 12.74586775183861

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