Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.0301 - 0.999i$
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 + (−0.717 + 2.54i)7-s + 0.999i·8-s + (5.09 − 2.94i)11-s + 4.05i·13-s + (−1.89 + 1.84i)14-s + (−0.5 + 0.866i)16-s + (−0.214 − 0.371i)17-s + (5.30 + 3.06i)19-s + 5.88·22-s + (−1.51 − 0.876i)23-s + (−2.02 + 3.51i)26-s + (−2.56 + 0.651i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.271 + 0.962i)7-s + 0.353i·8-s + (1.53 − 0.886i)11-s + 1.12i·13-s + (−0.506 + 0.493i)14-s + (−0.125 + 0.216i)16-s + (−0.0520 − 0.0901i)17-s + (1.21 + 0.703i)19-s + 1.25·22-s + (−0.316 − 0.182i)23-s + (−0.397 + 0.689i)26-s + (−0.484 + 0.123i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0301 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0301 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0301 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.0301 - 0.999i)$
$L(1)$  $\approx$  $2.848762869$
$L(\frac12)$  $\approx$  $2.848762869$
$L(\frac{3}{2})$   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 \( 1 + (0.717 - 2.54i)T \)
good11 \( 1 + (-5.09 + 2.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.05iT - 13T^{2} \)
17 \( 1 + (0.214 + 0.371i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.30 - 3.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 + 0.876i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 + 4.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.536 - 0.928i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.61T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + (0.481 - 0.834i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.3 + 6.57i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.77 - 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 0.609i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.32 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.54iT - 71T^{2} \)
73 \( 1 + (8.08 - 4.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.35 - 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (3.15 - 5.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.59iT - 97T^{2} \)
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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

−8.616274101026724297109885264871, −8.368616640299108247349971843231, −7.05207927233834351740921850604, −6.54892860549530745559194205372, −5.88487034426007766569831805900, −5.17494393128281956942040736765, −4.09782400452476513678505630936, −3.50841605793831662135717339754, −2.49007564090593724245891290990, −1.32987190260180329808622785417, 0.78425726676632368968361043060, 1.73139053788596888995434226819, 3.13302102474018256788588918555, 3.64122005588832429433033356947, 4.60210426339480536629599254010, 5.18939490931841129744386674169, 6.35436946423872255177001404946, 6.85917030044502003048837129176, 7.53688411287066103786858473208, 8.509443842234330445652174081516

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