Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.947 + 0.318i$
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 + (1.22 − 2.34i)7-s + 0.999·8-s + (2.03 + 1.17i)11-s + 4.64·13-s + (−2.64 + 0.107i)14-s + (−0.5 − 0.866i)16-s + (3.95 + 2.28i)17-s + (−0.491 + 0.283i)19-s − 2.35i·22-s + (2.91 + 5.04i)23-s + (−2.32 − 4.02i)26-s + (1.41 + 2.23i)28-s + 2.55i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.464 − 0.885i)7-s + 0.353·8-s + (0.615 + 0.355i)11-s + 1.28·13-s + (−0.706 + 0.0287i)14-s + (−0.125 − 0.216i)16-s + (0.958 + 0.553i)17-s + (−0.112 + 0.0650i)19-s − 0.502i·22-s + (0.607 + 1.05i)23-s + (−0.455 − 0.789i)26-s + (0.267 + 0.422i)28-s + 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.947 + 0.318i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.947 + 0.318i)$
$L(1)$  $\approx$  $1.823895297$
$L(\frac12)$  $\approx$  $1.823895297$
$L(\frac{3}{2})$   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 \( 1 + (-1.22 + 2.34i)T \)
good11 \( 1 + (-2.03 - 1.17i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.64T + 13T^{2} \)
17 \( 1 + (-3.95 - 2.28i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.491 - 0.283i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.91 - 5.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.02 - 4.63i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 - 6.57iT - 43T^{2} \)
47 \( 1 + (-5.46 + 3.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.07 - 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.67 - 2.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 + (-4.10 + 7.11i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.13 + 7.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.171iT - 83T^{2} \)
89 \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.779862141265275291813209780354, −7.88359259077660764373656342499, −7.39964203154834922077748062584, −6.45666891076406189892510582583, −5.57608707121633098318669609751, −4.52512261697244094795017165382, −3.79164397237114894484400015790, −3.15906223829513709386214962168, −1.59900980312979239121581668161, −1.13339618085043715105545895333, 0.798398098359646490677667434134, 1.91976158767918665684454998420, 3.14533172093556906165273120966, 4.10617706000736038059854953745, 5.13714625258175722144146682603, 5.77166264628486524075948175465, 6.41815988868692049191642416952, 7.24241695945796731946620715340, 8.121247774890651244025403662053, 8.712999390861225467547004121632

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