Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.705 - 0.708i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.34 − 1.22i)7-s + 0.999i·8-s + (2.03 + 1.17i)11-s + 4.64i·13-s + (2.64 − 0.107i)14-s + (−0.5 − 0.866i)16-s + (2.28 − 3.95i)17-s + (0.491 − 0.283i)19-s − 2.35·22-s + (−5.04 + 2.91i)23-s + (−2.32 − 4.02i)26-s + (−2.23 + 1.41i)28-s − 2.55i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.885 − 0.464i)7-s + 0.353i·8-s + (0.615 + 0.355i)11-s + 1.28i·13-s + (0.706 − 0.0287i)14-s + (−0.125 − 0.216i)16-s + (0.553 − 0.958i)17-s + (0.112 − 0.0650i)19-s − 0.502·22-s + (−1.05 + 0.607i)23-s + (−0.455 − 0.789i)26-s + (−0.422 + 0.267i)28-s − 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.705 - 0.708i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.705 - 0.708i)$
$L(1)$  $\approx$  $0.6188054093$
$L(\frac12)$  $\approx$  $0.6188054093$
$L(\frac{3}{2})$   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 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.34 + 1.22i)T \)
good11 \( 1 + (-2.03 - 1.17i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.64iT - 13T^{2} \)
17 \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.491 + 0.283i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.04 - 2.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.63 - 8.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + (3.15 + 5.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.5 + 6.07i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (-2.00 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 + (-7.11 - 4.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.13 - 7.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.171T + 83T^{2} \)
89 \( 1 + (2.72 + 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−9.237873783047053085251389950547, −8.094083131529514068828651128969, −7.49672472472026640661873033551, −6.60964809636196808771456174489, −6.37806603756571573228344138942, −5.22365241811826377764322361121, −4.27408693513286024886777837946, −3.46245305067011653677358439100, −2.27240291039242667920406478882, −1.15242760817457325700261456645, 0.25924861545441536925406195797, 1.53370053170960998076285950904, 2.75486832719398140029264912632, 3.40258196109789578269905198775, 4.26407550317351241295857695648, 5.69877711594075047317201285206, 6.03687616750445101992757701111, 6.95703814533600182007046882755, 7.961380576660756473583981280127, 8.323967857511365171762690629212

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