Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.579 + 0.815i$
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 + (2.61 − 0.397i)7-s + 0.999·8-s + (0.429 + 0.248i)11-s − 2.74·13-s + (−1.65 − 2.06i)14-s + (−0.5 − 0.866i)16-s + (3.16 + 1.82i)17-s + (−3.12 + 1.80i)19-s − 0.496i·22-s + (−3.21 − 5.56i)23-s + (1.37 + 2.37i)26-s + (−0.963 + 2.46i)28-s − 8.87i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.988 − 0.150i)7-s + 0.353·8-s + (0.129 + 0.0748i)11-s − 0.761·13-s + (−0.441 − 0.552i)14-s + (−0.125 − 0.216i)16-s + (0.768 + 0.443i)17-s + (−0.716 + 0.413i)19-s − 0.105i·22-s + (−0.669 − 1.16i)23-s + (0.269 + 0.466i)26-s + (−0.182 + 0.465i)28-s − 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.579 + 0.815i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3150,\ (\ :1/2),\ -0.579 + 0.815i)\)
\(L(1)\)  \(\approx\)  \(1.172850239\)
\(L(\frac12)\)  \(\approx\)  \(1.172850239\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.61 + 0.397i)T \)
good11 \( 1 + (-0.429 - 0.248i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.74T + 13T^{2} \)
17 \( 1 + (-3.16 - 1.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 - 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.21 + 5.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 + 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.98 - 1.14i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 + (-5.66 + 3.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.88 + 6.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.05 + 5.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.08 - 4.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (6.53 - 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.44 + 7.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.79iT - 83T^{2} \)
89 \( 1 + (0.743 + 1.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.05T + 97T^{2} \)
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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

−8.325617221980987450117899696602, −7.933031437949629232356087958660, −7.14899420074349837222187148618, −6.11364005756875293992082990570, −5.24512176733257976721971577642, −4.33160608890495408077788328544, −3.76265465112195692799344968406, −2.39766781930435472219514415862, −1.81693232569039083849698049398, −0.42942452162149267755133422479, 1.21628848267665165859492623707, 2.20143866198516728037186537988, 3.47479243070455586117822223866, 4.51294193437958122552198139655, 5.31357349921748541782527700442, 5.73701457070554383448548387707, 7.07982373908708093963910328519, 7.29410679870162252033707414634, 8.185268288888738195421116396932, 8.873847766703315298545815355262

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