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Properties

Label 2-303240-1.1-c1-0-54
Degree $2$
Conductor $303240$
Sign $-1$
Analytic cond. $2421.38$
Root an. cond. $49.2075$
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  + 3-s + 5-s + 7-s + 9-s + 4·11-s + 2·13-s + 15-s − 6·17-s + 21-s + 8·23-s + 25-s + 27-s + 2·29-s + 4·33-s + 35-s + 2·37-s + 2·39-s − 10·41-s + 4·43-s + 45-s + 49-s − 6·51-s − 14·53-s + 4·55-s − 12·59-s − 2·61-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.169·35-s + 0.328·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 1.92·53-s + 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(2421.38\)
Root analytic conductor: \(49.2075\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303240,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.91693673886286, −12.59309228117998, −11.97548043445171, −11.45579306124371, −11.00311594963683, −10.79045107680996, −10.12116823714539, −9.507324903750488, −9.112244854900582, −8.913572833715414, −8.408272651494755, −7.886167977382238, −7.267411848539850, −6.771090987880699, −6.393119384605128, −6.072938859315054, −5.147901429751173, −4.768140382861654, −4.381590135720246, −3.647367358855331, −3.282851756450568, −2.603638426863694, −2.053800368399653, −1.394478620064018, −1.090286042721310, 0, 1.090286042721310, 1.394478620064018, 2.053800368399653, 2.603638426863694, 3.282851756450568, 3.647367358855331, 4.381590135720246, 4.768140382861654, 5.147901429751173, 6.072938859315054, 6.393119384605128, 6.771090987880699, 7.267411848539850, 7.886167977382238, 8.408272651494755, 8.913572833715414, 9.112244854900582, 9.507324903750488, 10.12116823714539, 10.79045107680996, 11.00311594963683, 11.45579306124371, 11.97548043445171, 12.59309228117998, 12.91693673886286

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