Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.729 - 0.684i$
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 + (1.29 + 2.30i)7-s + 0.999i·8-s + (1.11 − 0.641i)11-s + 6.14i·13-s + (−0.0298 + 2.64i)14-s + (−0.5 + 0.866i)16-s + (3.26 + 5.64i)17-s + (−5.22 − 3.01i)19-s + 1.28·22-s + (−2.49 − 1.43i)23-s + (−3.07 + 5.32i)26-s + (−1.34 + 2.27i)28-s − 1.35i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.490 + 0.871i)7-s + 0.353i·8-s + (0.335 − 0.193i)11-s + 1.70i·13-s + (−0.00798 + 0.707i)14-s + (−0.125 + 0.216i)16-s + (0.790 + 1.37i)17-s + (−1.19 − 0.692i)19-s + 0.273·22-s + (−0.519 − 0.300i)23-s + (−0.602 + 1.04i)26-s + (−0.254 + 0.430i)28-s − 0.250i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.729 - 0.684i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.729 - 0.684i)$
$L(1)$  $\approx$  $2.381034518$
$L(\frac12)$  $\approx$  $2.381034518$
$L(\frac{3}{2})$   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 \( 1 + (-1.29 - 2.30i)T \)
good11 \( 1 + (-1.11 + 0.641i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.14iT - 13T^{2} \)
17 \( 1 + (-3.26 - 5.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.22 + 3.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.49 + 1.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.35iT - 29T^{2} \)
31 \( 1 + (7.49 - 4.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.76 + 8.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 + 5.35T + 43T^{2} \)
47 \( 1 + (-0.403 + 0.698i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.77 - 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.798 - 1.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.50 + 3.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.69 + 4.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (-10.7 + 6.20i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.59 + 9.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.74T + 83T^{2} \)
89 \( 1 + (1.81 - 3.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.76iT - 97T^{2} \)
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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

−8.955922656166479077893326488821, −8.189183468882877499486877401826, −7.43928262312082513283861271099, −6.38947091449975943676789644554, −6.11949607686388236135910807420, −5.13476047570412580357103795954, −4.29135464207215267502162790919, −3.71210085695875627100706060365, −2.36511563293239586738711800548, −1.70746580793223182734354475692, 0.56932397165040910442995847644, 1.66999972345748754845571306153, 2.86589223009908667134148019109, 3.65188188899768339814256491631, 4.46005466033195212876415535959, 5.25859882636262083070676171630, 5.94201817514295022415454289354, 6.87275343813319663675640652001, 7.78434144593459265119086602744, 8.057421013687131199151728608702

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