Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.997 - 0.0770i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.997 - 0.0770i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.997 - 0.0770i)$
$L(1)$  $\approx$  $0.4268494556$
$L(\frac12)$  $\approx$  $0.4268494556$
$L(\frac{3}{2})$   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

−8.325216072887096365601816161219, −7.69361714471615084625729181431, −6.44425115878494637636221493414, −5.84103669325253858493345612115, −5.43170881401605845147023612386, −4.25104168250149792897821934156, −3.56812767821076220240564918983, −2.52530278512396953549639681995, −1.85049687518661257144683085883, −0.094475653938951892950804924922, 1.66393858436422392698673956814, 2.69462950058557415519985492727, 3.78255674040632228824569730753, 4.57259074420352320391899188166, 4.89592625487516828904195554351, 6.36644708102963304482547213071, 6.73210556203100903110384389271, 7.24638479589134406138370944101, 8.304141191008703165006237868851, 8.923280914554081845716405123607

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