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Properties

Label 2-3528-168.53-c0-0-1
Degree $2$
Conductor $3528$
Sign $0.292 + 0.956i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.292 + 0.956i$
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: \((2,\ 3528,\ (\ :0),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268022806\)
\(L(\frac12)\) \(\approx\) \(1.268022806\)
\(L(1)\) 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

−8.780312959980162608173670742875, −8.010580278334672269543365228309, −7.11615039898288692902898762939, −6.28488932676919111322491695236, −5.26451341745514001322245435308, −4.67624658958807035023456434687, −4.00888531448077364603780936410, −2.75017158416967604220150262965, −2.32730633195062538784164847235, −0.939996646949474456777653101713, 1.00002213832800199030328525745, 2.93419759659996424304703521628, 3.33461052574574393921633214224, 4.66135020526321983001178050461, 5.08308037437324236599129716512, 6.06729946575061121346226634316, 6.49652696395894179705479279149, 7.52264159035691115995251557426, 8.023417646049248848518633372498, 8.720510872866618155134027026125

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