Properties

Label 2-3528-504.419-c0-0-5
Degree $2$
Conductor $3528$
Sign $0.873 + 0.487i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
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 + (0.991 − 0.130i)3-s + (0.499 − 0.866i)4-s + (−0.793 + 0.608i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (1.67 − 0.965i)11-s + (0.382 − 0.923i)12-s + (−0.5 − 0.866i)16-s − 1.58·17-s + (−0.707 + 0.707i)18-s − 1.98i·19-s + (−0.965 + 1.67i)22-s + (0.130 + 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.991 − 0.130i)3-s + (0.499 − 0.866i)4-s + (−0.793 + 0.608i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (1.67 − 0.965i)11-s + (0.382 − 0.923i)12-s + (−0.5 − 0.866i)16-s − 1.58·17-s + (−0.707 + 0.707i)18-s − 1.98i·19-s + (−0.965 + 1.67i)22-s + (0.130 + 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.873 + 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283362147\)
\(L(\frac12)\) \(\approx\) \(1.283362147\)
\(L(1)\) 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 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.58T + T^{2} \)
19 \( 1 + 1.98iT - T^{2} \)
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 \( 1 + (0.608 - 1.05i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.261iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 0.765T + T^{2} \)
97 \( 1 + (-1.05 + 0.608i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.787459154259770291149695535740, −8.216358255982566565717489578304, −7.14762817868011630627285677640, −6.68567065898247363001706577482, −6.16015746159464495207822800559, −4.80977284425169751501958580236, −4.06435557386923200013878353481, −2.90424761936545679746588780803, −2.06540145571525226022308601034, −0.921989437552608876905677201784, 1.70221607101990462068950577605, 1.88992426569151772008897104938, 3.30055307056773703768622877363, 3.92078110698986276406667974177, 4.55504392092807688854942092671, 6.16500156134731905488670324587, 6.88823694030901940640132238621, 7.49785564901663523390418107983, 8.256471311903879703470917787906, 8.988482611581605749545797110434

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