Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.994 - 0.106i$
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 + (−0.189 + 2.63i)7-s + 0.999i·8-s + (−2.55 + 1.47i)11-s + 3.93i·13-s + (−1.48 + 2.19i)14-s + (−0.5 + 0.866i)16-s + (−0.199 − 0.346i)17-s + (0.0305 + 0.0176i)19-s − 2.94·22-s + (3.23 + 1.86i)23-s + (−1.96 + 3.40i)26-s + (−2.38 + 1.15i)28-s − 8.89i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0716 + 0.997i)7-s + 0.353i·8-s + (−0.769 + 0.444i)11-s + 1.09i·13-s + (−0.396 + 0.585i)14-s + (−0.125 + 0.216i)16-s + (−0.0484 − 0.0839i)17-s + (0.00700 + 0.00404i)19-s − 0.628·22-s + (0.673 + 0.389i)23-s + (−0.385 + 0.667i)26-s + (−0.449 + 0.218i)28-s − 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.994 - 0.106i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.994 - 0.106i)$
$L(1)$  $\approx$  $1.557685623$
$L(\frac12)$  $\approx$  $1.557685623$
$L(\frac{3}{2})$   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 \( 1 + (0.189 - 2.63i)T \)
good11 \( 1 + (2.55 - 1.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.93iT - 13T^{2} \)
17 \( 1 + (0.199 + 0.346i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0305 - 0.0176i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 - 1.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-0.717 + 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.96 - 6.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 - 3.03T + 43T^{2} \)
47 \( 1 + (2.90 - 5.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.72 - 2.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.78 + 4.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.97 + 5.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.25 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 + (-0.297 + 0.171i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.15 - 7.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (-3.08 + 5.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.6iT - 97T^{2} \)
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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

−9.073001953704082155154597454111, −8.111663271499211310496404019194, −7.58403734467139884556234286361, −6.53432200817726472159501178911, −6.14263400695913073967087902883, −5.03732469405460327423565736245, −4.70860975191696333665313116928, −3.52605672019096578168114958964, −2.62606557595000936560621918606, −1.80075146859170239376575391977, 0.36426027656358802423075337262, 1.53602425015116208156085436635, 2.92616631411907260065898799591, 3.38553663913116955537356845714, 4.43004799157683683118238136598, 5.19870449834161559299212887900, 5.83624122421751621383979549828, 6.89070404813788554332598132649, 7.40918347353374272868713145835, 8.294630622782402262875312649870

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