Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.405 + 0.914i$
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 + (−0.295 + 2.62i)7-s + 0.999i·8-s + (0.570 + 0.329i)11-s − 6.13i·13-s + (−1.05 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (2.43 − 4.22i)17-s + (−6.30 + 3.63i)19-s − 0.659·22-s + (3.98 − 2.29i)23-s + (3.06 + 5.31i)26-s + (2.12 + 1.57i)28-s + 8.09i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.111 + 0.993i)7-s + 0.353i·8-s + (0.172 + 0.0993i)11-s − 1.70i·13-s + (−0.282 − 0.648i)14-s + (−0.125 − 0.216i)16-s + (0.591 − 1.02i)17-s + (−1.44 + 0.834i)19-s − 0.140·22-s + (0.830 − 0.479i)23-s + (0.601 + 1.04i)26-s + (0.402 + 0.296i)28-s + 1.50i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.405 + 0.914i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.405 + 0.914i)$
$L(1)$  $\approx$  $0.4232960123$
$L(\frac12)$  $\approx$  $0.4232960123$
$L(\frac{3}{2})$   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 \( 1 + (0.295 - 2.62i)T \)
good11 \( 1 + (-0.570 - 0.329i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.13iT - 13T^{2} \)
17 \( 1 + (-2.43 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.30 - 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.98 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + (-0.759 - 0.438i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.05 + 8.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 + 9.03T + 43T^{2} \)
47 \( 1 + (-6.00 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.5 + 6.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0618 - 0.0357i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.666 - 1.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.60iT - 71T^{2} \)
73 \( 1 + (2.44 + 1.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.88 + 5.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.44T + 83T^{2} \)
89 \( 1 + (2.66 + 4.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−8.457485978017494108577569329111, −7.86880508724888256674616447333, −6.97481380373459203076882031232, −6.25097906007067770610186735087, −5.40396730063998344474571808761, −4.96737902360356861960145303904, −3.44751482879048554965530372348, −2.71949394203775902153203276761, −1.57932205491744375333052614735, −0.16410630400230447268658499764, 1.28707100141906610248320099567, 2.12928458071290055951116132290, 3.40724678686088853342088150081, 4.12534373199700575686309201732, 4.82767651672071946812942265667, 6.28784012032760489314586102941, 6.71629263056520239680052705492, 7.41974970474554264149386414986, 8.401466029028478498959918675262, 8.815298495540326543516096131303

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