Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.904 + 0.426i$
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 + (−2.62 + 0.358i)7-s + 0.999i·8-s + (2.59 − 1.5i)11-s + 2.44i·13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (−2.95 − 5.12i)17-s + (−5.12 − 2.95i)19-s + 3·22-s + (3.67 + 2.12i)23-s + (−1.22 + 2.12i)26-s + (−1.62 − 2.09i)28-s + 7.24i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.990 + 0.135i)7-s + 0.353i·8-s + (0.783 − 0.452i)11-s + 0.679i·13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.717 − 1.24i)17-s + (−1.17 − 0.678i)19-s + 0.639·22-s + (0.766 + 0.442i)23-s + (−0.240 + 0.416i)26-s + (−0.306 − 0.395i)28-s + 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.904 + 0.426i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.904 + 0.426i)$
$L(1)$  $\approx$  $0.2222066651$
$L(\frac12)$  $\approx$  $0.2222066651$
$L(\frac{3}{2})$   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 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 + (2.95 - 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.27 - 3.62i)T + (26.5 - 45.8i)T^{2} \)
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 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−9.121459159640400261312471368670, −8.504535454979401157833398782855, −7.17701235332122296376303620189, −6.80900907234721353519048902625, −6.28137440783241413565822607429, −5.21164636103520679750776188143, −4.57564821752908913182945406227, −3.52730612071327029803041952518, −2.97194830648229878207283551245, −1.72737407971386825805044017987, 0.05048182065734429726723121525, 1.63844133290403571841484864048, 2.54948464887696042426060592854, 3.74688721352898580161200335320, 4.01082792997498821114859767426, 5.12416762124942128627366883664, 6.19364594951303207359476065715, 6.40915227687802008816452875804, 7.33827239047420274096571903256, 8.352794384127924307476846417370

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