Properties

Label 2-303450-1.1-c1-0-179
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 $2$

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 + 2·13-s + 14-s + 16-s − 18-s − 8·19-s + 21-s − 4·23-s + 24-s − 2·26-s − 27-s − 28-s − 6·29-s + 4·31-s − 32-s + 36-s − 10·37-s + 8·38-s − 2·39-s − 4·41-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.218·21-s − 0.834·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 1.29·38-s − 0.320·39-s − 0.624·41-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: \(2\)
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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.10098223614986, −12.57845933882579, −12.12122940472866, −11.75834239572636, −11.21764122360598, −10.66115578283675, −10.50978989801675, −10.03765939870985, −9.451616406891084, −9.004189250794884, −8.516233204812917, −8.172204713902175, −7.569257929227869, −7.027972301381833, −6.543416842692684, −6.264148218071806, −5.675605089581749, −5.280229267427041, −4.485640303371756, −3.976571092670515, −3.611701907998729, −2.789168900489424, −2.189000254090805, −1.688725098679849, −1.049075765306317, 0, 0, 1.049075765306317, 1.688725098679849, 2.189000254090805, 2.789168900489424, 3.611701907998729, 3.976571092670515, 4.485640303371756, 5.280229267427041, 5.675605089581749, 6.264148218071806, 6.543416842692684, 7.027972301381833, 7.569257929227869, 8.172204713902175, 8.516233204812917, 9.004189250794884, 9.451616406891084, 10.03765939870985, 10.50978989801675, 10.66115578283675, 11.21764122360598, 11.75834239572636, 12.12122940472866, 12.57845933882579, 13.10098223614986

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