Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.978 - 0.204i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + (1.80 + 1.93i)7-s + i·8-s − 3.87i·11-s − 1.60i·13-s + (1.93 − 1.80i)14-s + 16-s − 8.11·17-s − 2.63i·19-s − 3.87·22-s + 5.47i·23-s − 1.60·26-s + (−1.80 − 1.93i)28-s − 5.47i·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.681 + 0.732i)7-s + 0.353i·8-s − 1.16i·11-s − 0.445i·13-s + (0.517 − 0.481i)14-s + 0.250·16-s − 1.96·17-s − 0.605i·19-s − 0.825·22-s + 1.14i·23-s − 0.314·26-s + (−0.340 − 0.366i)28-s − 1.01i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.978 - 0.204i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.978 - 0.204i)$
$L(1)$  $\approx$  $0.5928019746$
$L(\frac12)$  $\approx$  $0.5928019746$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.80 - 1.93i)T \)
good11 \( 1 + 3.87iT - 11T^{2} \)
13 \( 1 + 1.60iT - 13T^{2} \)
17 \( 1 + 8.11T + 17T^{2} \)
19 \( 1 + 2.63iT - 19T^{2} \)
23 \( 1 - 5.47iT - 23T^{2} \)
29 \( 1 + 5.47iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
41 \( 1 + 1.60T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 2.26iT - 53T^{2} \)
59 \( 1 - 4.61T + 59T^{2} \)
61 \( 1 + 11.8iT - 61T^{2} \)
67 \( 1 + 6.90T + 67T^{2} \)
71 \( 1 + 2.63iT - 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + 8.01T + 79T^{2} \)
83 \( 1 + 3.20T + 83T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 - 8.68iT - 97T^{2} \)
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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

−8.497751271920286767905458231942, −7.79869104699280922849865508559, −6.69557778184902040666056458348, −5.85857800286351492991808867492, −5.12124781545003139721157885657, −4.36304818353270871408320408356, −3.32997362348800033108465860902, −2.51523141336406952461333943888, −1.60369212421406493496746736545, −0.17738316155944236902699729960, 1.48159398043644526969639088222, 2.47447128797894892784832642815, 4.08469112682378284418297515869, 4.38988686497188581273259350441, 5.16006982357308299414547106856, 6.25679117146348184913465837860, 7.01505414955916984231233224994, 7.32230419111969905971812361048, 8.393900552898553229663129321660, 8.808520902915412005997042444467

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