Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.935 - 0.354i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.358 + 2.62i)7-s + 0.999·8-s + (−2.59 + 1.5i)11-s − 2.44·13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (5.12 − 2.95i)17-s + (5.12 + 2.95i)19-s − 3i·22-s + (2.12 − 3.67i)23-s + (1.22 − 2.12i)26-s + (2.09 − 1.62i)28-s + 7.24i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.135 + 0.990i)7-s + 0.353·8-s + (−0.783 + 0.452i)11-s − 0.679·13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (1.24 − 0.717i)17-s + (1.17 + 0.678i)19-s − 0.639i·22-s + (0.442 − 0.766i)23-s + (0.240 − 0.416i)26-s + (0.395 − 0.306i)28-s + 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.935 - 0.354i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.935 - 0.354i)$
$L(1)$  $\approx$  $0.9971181338$
$L(\frac12)$  $\approx$  $0.9971181338$
$L(\frac{3}{2})$   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 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.358 - 2.62i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + (-5.12 + 2.95i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.12 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.210 - 0.121i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 0.242iT - 43T^{2} \)
47 \( 1 + (5.12 + 2.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.62 - 6.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.03 + 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (-0.717 - 1.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−9.061191014325594727237089161496, −8.155326749511602659256105027257, −7.49582240599904201731707524539, −7.02446640263090839683550234728, −5.81113433686617261009865356083, −5.32411883628217648335473301347, −4.77913407573857216977377409521, −3.33179311236744859505928227049, −2.51364843617247239667898348552, −1.28141558610562689170863959346, 0.37980302078076454295286044726, 1.42381403937196745603907718074, 2.67182802459757513713878044661, 3.45914942624014748654530272389, 4.27836680682684524749952551898, 5.24596379436587874517004506483, 5.94042373043440272690822559502, 7.34600889313453639597071753036, 7.55185508136405821109247193417, 8.234833127675441162789077524117

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