Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2} $
Sign $1$
Motivic weight 1
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 + 12-s − 2·13-s + 14-s + 16-s + 18-s + 8·19-s + 21-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 + 6·41-s + 42-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 + 0.204·24-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 + 0.937·41-s + 0.154·42-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

\( d \)  =  \(2\)
\( N \)  =  \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{303450} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 303450,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.126097427\)
\(L(\frac12)\)  \(\approx\)  \(6.126097427\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 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 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.84149685339839, −12.10068735175593, −11.91568378032180, −11.47226999749552, −10.87601090579392, −10.46018486510765, −9.828312915421131, −9.588650523666196, −8.950086029396467, −8.547348976398977, −7.875574951288122, −7.514827347384450, −7.117989568233809, −6.758189327136887, −5.779250470159955, −5.626746721282268, −5.098188577351564, −4.459541061603764, −4.096024393978286, −3.455375803592424, −2.922752647729670, −2.598064998785853, −1.748417844913970, −1.400334743253242, −0.5372615370725890, 0.5372615370725890, 1.400334743253242, 1.748417844913970, 2.598064998785853, 2.922752647729670, 3.455375803592424, 4.096024393978286, 4.459541061603764, 5.098188577351564, 5.626746721282268, 5.779250470159955, 6.758189327136887, 7.117989568233809, 7.514827347384450, 7.875574951288122, 8.547348976398977, 8.950086029396467, 9.588650523666196, 9.828312915421131, 10.46018486510765, 10.87601090579392, 11.47226999749552, 11.91568378032180, 12.10068735175593, 12.84149685339839

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