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Properties

Label 2-3528-392.101-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.518 + 0.855i$
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  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.246 − 0.167i)5-s + (0.826 − 0.563i)7-s + (−0.433 − 0.900i)8-s + (0.167 + 0.246i)10-s + (−0.218 − 1.44i)11-s + (−0.974 + 0.222i)14-s + (0.0747 + 0.997i)16-s + (−0.0663 − 0.290i)20-s + (−0.326 + 1.42i)22-s + (−0.332 − 0.848i)25-s + (0.988 + 0.149i)28-s + (0.712 − 0.162i)29-s + (−1.72 − 0.997i)31-s + (0.294 − 0.955i)32-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.246 − 0.167i)5-s + (0.826 − 0.563i)7-s + (−0.433 − 0.900i)8-s + (0.167 + 0.246i)10-s + (−0.218 − 1.44i)11-s + (−0.974 + 0.222i)14-s + (0.0747 + 0.997i)16-s + (−0.0663 − 0.290i)20-s + (−0.326 + 1.42i)22-s + (−0.332 − 0.848i)25-s + (0.988 + 0.149i)28-s + (0.712 − 0.162i)29-s + (−1.72 − 0.997i)31-s + (0.294 − 0.955i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2845, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7022305454\)
\(L(\frac12)\) \(\approx\) \(0.7022305454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.930 + 0.365i)T \)
3 \( 1 \)
7 \( 1 + (-0.826 + 0.563i)T \)
good5 \( 1 + (0.246 + 0.167i)T + (0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.218 + 1.44i)T + (-0.955 + 0.294i)T^{2} \)
13 \( 1 + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.712 + 0.162i)T + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (0.680 - 0.733i)T + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (1.43 - 0.975i)T + (0.365 - 0.930i)T^{2} \)
61 \( 1 + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.807 - 0.317i)T + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.16 + 1.45i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + 0.589iT - T^{2} \)
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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

−8.507251230403708685347956144261, −7.87035203546041289329346223783, −7.40695373045765272049076221881, −6.35652657330505416643805441444, −5.64992274500464997081011295747, −4.46792795331858971837543791131, −3.69989056074520722769045800394, −2.78791680394670560968213178627, −1.67739150551778572664969231956, −0.55892190719950638605345557258, 1.56680934053516157428729409084, 2.18412842740485329509249718573, 3.37751903222506865047405308878, 4.76250691891918861348296793220, 5.19768922526796257647685769969, 6.18324519751974035272498284990, 7.05511323297420326431689109077, 7.57927788447719265587710465047, 8.166327893276351526835102855561, 9.067890138095837356959723776300

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