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Properties

Label 2-303450-1.1-c1-0-151
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 − 2·11-s − 12-s + 2·13-s + 14-s + 16-s + 18-s − 4·19-s − 21-s − 2·22-s + 2·23-s − 24-s + 2·26-s − 27-s + 28-s − 2·29-s + 8·31-s + 32-s + 2·33-s + 36-s + 6·37-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 − 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.426·22-s + 0.417·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s + 0.986·37-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.91376050759047, −12.51479268045266, −12.05923921146749, −11.48291144783776, −11.04464852044370, −10.87027932134501, −10.39910766255801, −9.622434722935441, −9.497002734631624, −8.586933393500582, −8.158587365961939, −7.857018691742184, −7.181950655480054, −6.717620529743629, −6.134974073264649, −5.963397223188616, −5.218141699951934, −4.905270986734219, −4.265457049690853, −4.030906882948529, −3.235284526776800, −2.622205331527223, −2.225326688994692, −1.386172271228709, −0.8966038632020374, 0, 0.8966038632020374, 1.386172271228709, 2.225326688994692, 2.622205331527223, 3.235284526776800, 4.030906882948529, 4.265457049690853, 4.905270986734219, 5.218141699951934, 5.963397223188616, 6.134974073264649, 6.717620529743629, 7.181950655480054, 7.857018691742184, 8.158587365961939, 8.586933393500582, 9.497002734631624, 9.622434722935441, 10.39910766255801, 10.87027932134501, 11.04464852044370, 11.48291144783776, 12.05923921146749, 12.51479268045266, 12.91376050759047

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