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Properties

Label 2-3528-504.227-c0-0-6
Degree $2$
Conductor $3528$
Sign $-0.810 - 0.585i$
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  − i·2-s + (−0.991 + 0.130i)3-s − 4-s + (0.130 + 0.991i)6-s + i·8-s + (0.965 − 0.258i)9-s + (−0.448 + 0.258i)11-s + (0.991 − 0.130i)12-s + 16-s + (−0.130 + 0.226i)17-s + (−0.258 − 0.965i)18-s + (−1.05 + 0.608i)19-s + (0.258 + 0.448i)22-s + (−0.130 − 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  − i·2-s + (−0.991 + 0.130i)3-s − 4-s + (0.130 + 0.991i)6-s + i·8-s + (0.965 − 0.258i)9-s + (−0.448 + 0.258i)11-s + (0.991 − 0.130i)12-s + 16-s + (−0.130 + 0.226i)17-s + (−0.258 − 0.965i)18-s + (−1.05 + 0.608i)19-s + (0.258 + 0.448i)22-s + (−0.130 − 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1498205149\)
\(L(\frac12)\) \(\approx\) \(0.1498205149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.991 - 0.130i)T \)
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 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.05 - 0.608i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + 1.58T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.71 + 0.991i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.466325491529315688470675358563, −7.65217675436742055340106192736, −6.70292357865923757871992198924, −5.80688932972307074953184513416, −5.23092642721325930450720518196, −4.27113274782639217526329698178, −3.82974254700217258131185546546, −2.47554550844879415462250614906, −1.60178836186570864769219561240, −0.10446153697713211463722266382, 1.42044875001478911962576463015, 3.00897893166510766866722333171, 4.26353908103236754796489343948, 4.77651982881301829085081705945, 5.58969869115737085196320425462, 6.25770591153626166472649102866, 6.80351075780713892643269349230, 7.65610440197736046509113911970, 8.165251209634277122022697477664, 9.197729127709210505944461577270

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