Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.972 - 0.231i$
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 + (−2.63 − 0.189i)7-s − 0.999i·8-s + (−1.32 + 0.766i)11-s − 1.48i·13-s + (2.19 + 1.48i)14-s + (−0.5 + 0.866i)16-s + (−1.21 − 2.10i)17-s + (4.21 + 2.43i)19-s + 1.53·22-s + (−0.232 − 0.133i)23-s + (−0.741 + 1.28i)26-s + (−1.15 − 2.38i)28-s − 0.898i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.997 − 0.0716i)7-s − 0.353i·8-s + (−0.400 + 0.230i)11-s − 0.411i·13-s + (0.585 + 0.396i)14-s + (−0.125 + 0.216i)16-s + (−0.294 − 0.510i)17-s + (0.966 + 0.557i)19-s + 0.326·22-s + (−0.0483 − 0.0279i)23-s + (−0.145 + 0.251i)26-s + (−0.218 − 0.449i)28-s − 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.972 - 0.231i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.972 - 0.231i)$
$L(1)$  $\approx$  $0.8932038645$
$L(\frac12)$  $\approx$  $0.8932038645$
$L(\frac{3}{2})$   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 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + (1.32 - 0.766i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.48iT - 13T^{2} \)
17 \( 1 + (1.21 + 2.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.21 - 2.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.232 + 0.133i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 - 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.74 - 4.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + (-3.72 + 6.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.00 - 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.12 - 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.73 - 3.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.01 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (0.297 - 0.171i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.22 + 9.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.45T + 83T^{2} \)
89 \( 1 + (-7.98 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−8.827672273239019027916524870453, −8.015374561841322692668432608461, −7.26843226774901788863281787642, −6.66315953514618056140999038839, −5.72368190297401210688259578951, −4.88728593197503971997627756759, −3.66695720721281598193978033245, −3.07970134418292686914415470137, −2.08869691299227641257062703458, −0.74327576339253188545193901259, 0.50037622870713479249850232395, 1.91416895571848540848711294336, 2.97926145368555365668444617717, 3.82680308025975049755083578729, 5.03290351529396759226411224861, 5.72717148729081581149632031585, 6.62893124815370870437042260560, 7.03622396840272655431755285704, 7.985667786645284736492851130593, 8.653892932769179852708956284828

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