Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.106 + 0.994i$
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 + (2.63 − 0.189i)7-s + 0.999i·8-s + (−4.67 − 2.69i)11-s + 2.51i·13-s + (−2.19 + 1.48i)14-s + (−0.5 − 0.866i)16-s + (−2.24 + 3.89i)17-s + (−2.48 + 1.43i)19-s + 5.39·22-s + (−0.232 + 0.133i)23-s + (−1.25 − 2.18i)26-s + (1.15 − 2.38i)28-s − 8.89i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.997 − 0.0716i)7-s + 0.353i·8-s + (−1.40 − 0.813i)11-s + 0.698i·13-s + (−0.585 + 0.396i)14-s + (−0.125 − 0.216i)16-s + (−0.545 + 0.945i)17-s + (−0.568 + 0.328i)19-s + 1.15·22-s + (−0.0483 + 0.0279i)23-s + (−0.246 − 0.427i)26-s + (0.218 − 0.449i)28-s − 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.106 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.106 + 0.994i)$
$L(1)$  $\approx$  $0.7909116008$
$L(\frac12)$  $\approx$  $0.7909116008$
$L(\frac{3}{2})$   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 \( 1 + (-2.63 + 0.189i)T \)
good11 \( 1 + (4.67 + 2.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.51iT - 13T^{2} \)
17 \( 1 + (2.24 - 3.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.48 - 1.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.232 - 0.133i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-4.18 - 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.25 + 5.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.760T + 41T^{2} \)
43 \( 1 - 5.86T + 43T^{2} \)
47 \( 1 + (3.99 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.27 - 4.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.33 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.27 - 1.31i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (10.0 + 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.29 + 7.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 + (3.98 + 6.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.16iT - 97T^{2} \)
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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

−8.353110759853428150620417905844, −7.993330919766399777393910222762, −7.18384607668798252804751819784, −6.21604104321865591373854650844, −5.62358088289317983078406380346, −4.72804545747097940457104925476, −3.90864775175476760588257050990, −2.50865997447267313464070974330, −1.77156187178959820831441426959, −0.31615537077247545733565601265, 1.13953809679200660032134813389, 2.35992896738636617296814834189, 2.85392276138053913478938824054, 4.30026062852617539503413185818, 4.96300234014806645539184851180, 5.67197605585838601272716864827, 7.04701007064568274190110724719, 7.35689747364251707142645449645, 8.353709885382156559389614093613, 8.599733382363181860233269203542

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