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Properties

Label 2-3528-392.213-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.788 - 0.615i$
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  + (0.997 − 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.825 + 0.766i)5-s + (−0.733 − 0.680i)7-s + (0.974 − 0.222i)8-s + (−0.766 + 0.825i)10-s + (1.11 + 1.63i)11-s + (−0.781 − 0.623i)14-s + (0.955 − 0.294i)16-s + (−0.702 + 0.880i)20-s + (1.23 + 1.54i)22-s + (0.0201 − 0.268i)25-s + (−0.826 − 0.563i)28-s + (0.116 + 0.0931i)29-s + (0.510 + 0.294i)31-s + (0.930 − 0.365i)32-s + ⋯
L(s)  = 1  + (0.997 − 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.825 + 0.766i)5-s + (−0.733 − 0.680i)7-s + (0.974 − 0.222i)8-s + (−0.766 + 0.825i)10-s + (1.11 + 1.63i)11-s + (−0.781 − 0.623i)14-s + (0.955 − 0.294i)16-s + (−0.702 + 0.880i)20-s + (1.23 + 1.54i)22-s + (0.0201 − 0.268i)25-s + (−0.826 − 0.563i)28-s + (0.116 + 0.0931i)29-s + (0.510 + 0.294i)31-s + (0.930 − 0.365i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.105515318\)
\(L(\frac12)\) \(\approx\) \(2.105515318\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.997 + 0.0747i)T \)
3 \( 1 \)
7 \( 1 + (0.733 + 0.680i)T \)
good5 \( 1 + (0.825 - 0.766i)T + (0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-1.11 - 1.63i)T + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.116 - 0.0931i)T + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.510 - 0.294i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (-0.149 - 0.988i)T + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (-1.26 - 1.17i)T + (0.0747 + 0.997i)T^{2} \)
61 \( 1 + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.94 + 0.145i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.79 + 0.865i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + 1.86iT - T^{2} \)
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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

−8.843947598194040791394775719520, −7.49666487044696192158113835481, −7.28338948522420122610112475840, −6.66726140930846134133902009964, −5.97690164451989227006618940915, −4.68563616761213853721010363379, −4.17340221390434879691451090073, −3.51015123995511091501622032522, −2.69027763162928738560721810900, −1.47534586839237768646352364996, 0.977391265255056606794434867000, 2.45453693582586641840828146992, 3.48225128708273524195007640484, 3.86341288775342664691292271744, 4.85526978225777197215919971918, 5.67785715939557606966648591398, 6.29507721228526641345745720112, 6.91522045006171597375656214513, 8.067709363480762655088667797904, 8.493032819999720893079756722829

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