Properties

Degree $2$
Conductor $3528$
Sign $-0.997 + 0.0698i$
Motivic weight $0$
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 + (−0.793 − 0.608i)3-s + (0.499 − 0.866i)4-s + (−0.991 − 0.130i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−0.448 + 0.258i)11-s + (−0.923 + 0.382i)12-s + (−0.5 − 0.866i)16-s − 1.98·17-s + (0.707 + 0.707i)18-s − 1.58i·19-s + (−0.258 + 0.448i)22-s + (−0.608 + 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.793 − 0.608i)3-s + (0.499 − 0.866i)4-s + (−0.991 − 0.130i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−0.448 + 0.258i)11-s + (−0.923 + 0.382i)12-s + (−0.5 − 0.866i)16-s − 1.98·17-s + (0.707 + 0.707i)18-s − 1.58i·19-s + (−0.258 + 0.448i)22-s + (−0.608 + 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.997 + 0.0698i$
Motivic weight: \(0\)
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.997 + 0.0698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077029922\)
\(L(\frac12)\) \(\approx\) \(1.077029922\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.98T + T^{2} \)
19 \( 1 + 1.58iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.21iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + (0.226 - 0.130i)T + (0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.339040596568676480541981882405, −7.26254755566190090438517479398, −6.69041649964851577070800647531, −6.22425227832006943817623980599, −5.06095411439631144196945091071, −4.83276508774292458981608391775, −3.84273887698934728590329264334, −2.46165484513732746269842100542, −2.05049025907797447877295683635, −0.47599565512967195702798966557, 1.86501440951890565720959198219, 3.11346595661692576914119106311, 3.95425702767837524542230511472, 4.58268017700945694367887500753, 5.36958661709534005077741266073, 6.02950680851332723397946669553, 6.60826962574115090898007349615, 7.42679083627519369276291094154, 8.292881696418312391619742353075, 8.998533124361548697676237513612

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