Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.982 - 0.184i$
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 + (−0.397 + 2.61i)7-s + 0.999i·8-s + (−0.429 + 0.248i)11-s + 2.74i·13-s + (−1.65 + 2.06i)14-s + (−0.5 + 0.866i)16-s + (−1.82 − 3.16i)17-s + (3.12 + 1.80i)19-s − 0.496·22-s + (−5.56 − 3.21i)23-s + (−1.37 + 2.37i)26-s + (−2.46 + 0.963i)28-s + 8.87i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.150 + 0.988i)7-s + 0.353i·8-s + (−0.129 + 0.0748i)11-s + 0.761i·13-s + (−0.441 + 0.552i)14-s + (−0.125 + 0.216i)16-s + (−0.443 − 0.768i)17-s + (0.716 + 0.413i)19-s − 0.105·22-s + (−1.16 − 0.669i)23-s + (−0.269 + 0.466i)26-s + (−0.465 + 0.182i)28-s + 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.982 - 0.184i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.982 - 0.184i)$
$L(1)$  $\approx$  $1.517332938$
$L(\frac12)$  $\approx$  $1.517332938$
$L(\frac{3}{2})$   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 \( 1 + (0.397 - 2.61i)T \)
good11 \( 1 + (0.429 - 0.248i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.74iT - 13T^{2} \)
17 \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.12 - 1.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.56 + 3.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.14 + 1.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 + 2.22T + 43T^{2} \)
47 \( 1 + (-3.27 + 5.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.72 - 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.05 - 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 - 7.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (11.3 - 6.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.44 + 7.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 + (0.743 - 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−8.950714408567251627361670585587, −8.322850284993892509326145923028, −7.30730759834801918919169754045, −6.79992484689924757003397691521, −5.86931862572420638505299075277, −5.31022612462524249250983841638, −4.50009197302818381115809974634, −3.54392924968609011336527917572, −2.65659321830896814198367223125, −1.73782580204204265195285054249, 0.35073582075330024398336380611, 1.62690378792779794295874513050, 2.72291148817502723921518028824, 3.73835060132582215467498979955, 4.18864070049276288907085293729, 5.24555640453510024352953884642, 5.96968752203564972274981618876, 6.69306561010842476225781222073, 7.71057480866506914501408207618, 8.016072969135942521539505143592

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