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Properties

Label 2-303450-1.1-c1-0-0
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 − 3·11-s − 12-s + 3·13-s − 14-s + 16-s + 18-s + 6·19-s + 21-s − 3·22-s − 2·23-s − 24-s + 3·26-s − 27-s − 28-s − 6·29-s − 4·31-s + 32-s + 3·33-s + 36-s − 11·37-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 − 0.904·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.218·21-s − 0.639·22-s − 0.417·23-s − 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.80·37-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

\(L(1)\) \(\approx\) \(0.6262334496\)
\(L(\frac12)\) \(\approx\) \(0.6262334496\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−12.74955716394730, −12.16567641993339, −11.84482061954698, −11.33282003624452, −11.00924678893329, −10.43368635693653, −10.05955969605328, −9.635380619900909, −9.040067523227504, −8.374225095685895, −8.039306363964241, −7.329173990045111, −7.057437808178160, −6.496741655163624, −6.015518803080106, −5.435385005934465, −5.175027828425555, −4.805145489278334, −3.887072858697801, −3.474011235230499, −3.239824846143123, −2.400085231774757, −1.664533650159102, −1.337516171316130, −0.1841527240179104, 0.1841527240179104, 1.337516171316130, 1.664533650159102, 2.400085231774757, 3.239824846143123, 3.474011235230499, 3.887072858697801, 4.805145489278334, 5.175027828425555, 5.435385005934465, 6.015518803080106, 6.496741655163624, 7.057437808178160, 7.329173990045111, 8.039306363964241, 8.374225095685895, 9.040067523227504, 9.635380619900909, 10.05955969605328, 10.43368635693653, 11.00924678893329, 11.33282003624452, 11.84482061954698, 12.16567641993339, 12.74955716394730

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