Properties

Label 2-303450-1.1-c1-0-12
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 − 2·13-s + 14-s + 16-s − 18-s + 21-s + 22-s + 23-s + 24-s + 2·26-s − 27-s − 28-s + 6·29-s + 8·31-s − 32-s + 33-s + 36-s − 3·37-s + 2·39-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 − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s − 0.493·37-s + 0.320·39-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5683867589\)
\(L(\frac12)\) \(\approx\) \(0.5683867589\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.48831771889781, −12.11560317403345, −11.81915329609234, −11.23931097969159, −10.79135218183762, −10.25074144351872, −10.04941073387295, −9.585374082368026, −8.994244935870408, −8.541883398028296, −8.090064898902655, −7.525633700406067, −7.074939822503934, −6.684475771779887, −6.131586718181720, −5.728238164706924, −5.152524944045615, −4.526744873766388, −4.214447007901132, −3.320625871678033, −2.755608458284664, −2.459206328388414, −1.499842471547877, −1.061224095879830, −0.2614782855187145, 0.2614782855187145, 1.061224095879830, 1.499842471547877, 2.459206328388414, 2.755608458284664, 3.320625871678033, 4.214447007901132, 4.526744873766388, 5.152524944045615, 5.728238164706924, 6.131586718181720, 6.684475771779887, 7.074939822503934, 7.525633700406067, 8.090064898902655, 8.541883398028296, 8.994244935870408, 9.585374082368026, 10.04941073387295, 10.25074144351872, 10.79135218183762, 11.23931097969159, 11.81915329609234, 12.11560317403345, 12.48831771889781

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