Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.841 + 0.539i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.189 + 2.63i)7-s − 0.999·8-s + (1.32 − 0.766i)11-s + 1.48·13-s + (2.19 + 1.48i)14-s + (−0.5 + 0.866i)16-s + (2.10 − 1.21i)17-s + (−4.21 − 2.43i)19-s − 1.53i·22-s + (−0.133 + 0.232i)23-s + (0.741 − 1.28i)26-s + (2.38 − 1.15i)28-s − 0.898i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0716 + 0.997i)7-s − 0.353·8-s + (0.400 − 0.230i)11-s + 0.411·13-s + (0.585 + 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.510 − 0.294i)17-s + (−0.966 − 0.557i)19-s − 0.326i·22-s + (−0.0279 + 0.0483i)23-s + (0.145 − 0.251i)26-s + (0.449 − 0.218i)28-s − 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.841 + 0.539i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.841 + 0.539i)$
$L(1)$  $\approx$  $2.184087333$
$L(\frac12)$  $\approx$  $2.184087333$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.189 - 2.63i)T \)
good11 \( 1 + (-1.32 + 0.766i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.48T + 13T^{2} \)
17 \( 1 + (-2.10 + 1.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.21 + 2.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.133 - 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 - 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.74 - 2.74i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.76T + 41T^{2} \)
43 \( 1 + 1.86iT - 43T^{2} \)
47 \( 1 + (-6.46 - 3.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 - 3.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.73 - 3.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.8 + 8.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (0.171 + 0.297i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.22 - 9.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.45iT - 83T^{2} \)
89 \( 1 + (-7.98 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−8.804083851802739087539403944689, −8.056402613946099635136608126589, −6.99175390524715619949133128491, −6.09400424712966557195602149855, −5.62505848482766495809944176524, −4.64154489888308330218762717479, −3.86342734038313911886878015440, −2.86271381076813712738251757732, −2.17078393214648202450728531118, −0.892418335263318180479986101777, 0.830394166275732045472926204401, 2.17912690085035718951224898541, 3.58095200444671939617320762800, 4.00922321037532554325973583955, 4.86009561459342601398305214800, 5.88474844234640279158677222547, 6.45140459468676125294958788459, 7.23712782094807992514292566055, 7.87275274543956711893539395376, 8.560959950872013799728611736782

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