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Properties

Label 2-3528-168.53-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.999 + 0.0431i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (0.999 + i)22-s + (−1.22 + 0.707i)23-s + (0.5 − 0.866i)25-s − 1.41·29-s + (−0.965 + 0.258i)32-s + (1.73 − i)37-s − 2i·43-s + (−1.22 + 0.707i)44-s + (−0.366 − 1.36i)46-s + (0.707 + 0.707i)50-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (0.707 − 1.22i)11-s + (0.500 + 0.866i)16-s + (0.999 + i)22-s + (−1.22 + 0.707i)23-s + (0.5 − 0.866i)25-s − 1.41·29-s + (−0.965 + 0.258i)32-s + (1.73 − i)37-s − 2i·43-s + (−1.22 + 0.707i)44-s + (−0.366 − 1.36i)46-s + (0.707 + 0.707i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0431i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.999 + 0.0431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9474716070\)
\(L(\frac12)\) \(\approx\) \(0.9474716070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.667626543047344639019427950434, −7.989792323027937304679112027586, −7.32058494119961381897214944638, −6.43952259946837774407567928247, −5.88019465200607651246138994482, −5.25448924137894579268529384981, −4.07168668798910692651758241388, −3.61152388758837936762585376598, −2.06968480855519932043285812939, −0.67929544759419335869109956091, 1.28939132979973133206496192503, 2.16623689542713463482722731558, 3.11411195298631411564507029114, 4.18587757288400233761845006585, 4.54453568011987897141476413053, 5.64661979663595573825819163323, 6.59182342525945335544679366109, 7.52751631089730166670297889599, 8.042307530653917996097450556342, 9.049051712639536044116406067733

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