Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.425 + 0.904i$
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 + (−2.43 − 1.04i)7-s − 0.999i·8-s + (1.38 − 0.800i)11-s + 0.770i·13-s + (1.58 + 2.11i)14-s + (−0.5 + 0.866i)16-s + (−1.76 − 3.05i)17-s + (3.06 + 1.77i)19-s − 1.60·22-s + (2.79 + 1.61i)23-s + (0.385 − 0.667i)26-s + (−0.313 − 2.62i)28-s + 0.700i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.919 − 0.393i)7-s − 0.353i·8-s + (0.417 − 0.241i)11-s + 0.213i·13-s + (0.423 + 0.566i)14-s + (−0.125 + 0.216i)16-s + (−0.427 − 0.739i)17-s + (0.703 + 0.406i)19-s − 0.341·22-s + (0.582 + 0.336i)23-s + (0.0755 − 0.130i)26-s + (−0.0592 − 0.496i)28-s + 0.130i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.425 + 0.904i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.425 + 0.904i)$
$L(1)$  $\approx$  $0.8446171161$
$L(\frac12)$  $\approx$  $0.8446171161$
$L(\frac{3}{2})$   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 \( 1 + (2.43 + 1.04i)T \)
good11 \( 1 + (-1.38 + 0.800i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.770iT - 13T^{2} \)
17 \( 1 + (1.76 + 3.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.06 - 1.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.79 - 1.61i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.700iT - 29T^{2} \)
31 \( 1 + (-1.13 + 0.656i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.457 - 0.792i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 + 9.26T + 43T^{2} \)
47 \( 1 + (-1.33 + 2.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.04 + 4.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.56 - 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.43 + 5.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.40 + 5.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.47iT - 71T^{2} \)
73 \( 1 + (9.55 - 5.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.45 - 2.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 + (-4.40 + 7.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.31iT - 97T^{2} \)
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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

−8.633613805754469264196993896115, −7.67258488117097908457191283447, −7.01668202162919108595112507223, −6.41984828946391491582424872901, −5.45250291601673620660094860303, −4.37501763589950202425000588679, −3.46430520188639788704809857345, −2.81919845786051372607685424336, −1.53960566663728454937104995927, −0.37784477096074581555894883386, 1.05327321453389578440175640645, 2.34013123425712973000821783692, 3.23394980520095814286554096071, 4.28042810818161162453690626222, 5.30457301064666535556866565848, 6.08931391009571178901905755527, 6.71562994029466128014547689334, 7.36852628752070611957331041168, 8.269142922670266869907632169317, 9.023810653949147302628082445025

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