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Properties

Label 2-303450-1.1-c1-0-114
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 $1$

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 − 12-s + 6·13-s + 14-s + 16-s − 18-s + 21-s − 8·23-s + 24-s − 6·26-s − 27-s − 28-s + 6·29-s + 8·31-s − 32-s + 36-s + 10·37-s − 6·39-s + 6·41-s − 42-s − 12·43-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 − 0.288·12-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s − 1.66·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1/6·36-s + 1.64·37-s − 0.960·39-s + 0.937·41-s − 0.154·42-s − 1.82·43-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: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.83731881378264, −12.26531606216400, −11.87194651594737, −11.54770826247522, −11.04587709043131, −10.54411229888461, −10.12818679587148, −9.873492156302900, −9.249412203934037, −8.680775428086555, −8.344697486902779, −7.877311288544907, −7.426982611986142, −6.662429291100660, −6.280905164221383, −6.088419094354520, −5.599712448132477, −4.774399510059998, −4.221891445488133, −3.880533597889472, −3.037496888572007, −2.706778764467718, −1.831655256457648, −1.275408142236448, −0.7688815441957915, 0, 0.7688815441957915, 1.275408142236448, 1.831655256457648, 2.706778764467718, 3.037496888572007, 3.880533597889472, 4.221891445488133, 4.774399510059998, 5.599712448132477, 6.088419094354520, 6.280905164221383, 6.662429291100660, 7.426982611986142, 7.877311288544907, 8.344697486902779, 8.680775428086555, 9.249412203934037, 9.873492156302900, 10.12818679587148, 10.54411229888461, 11.04587709043131, 11.54770826247522, 11.87194651594737, 12.26531606216400, 12.83731881378264

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