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Properties

Label 2-3528-7.4-c1-0-17
Degree $2$
Conductor $3528$
Sign $-0.266 - 0.963i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)5-s + (1 − 1.73i)17-s + (−1 − 1.73i)19-s + (4 + 6.92i)23-s + (−5.49 + 9.52i)25-s − 2·29-s + (2 − 3.46i)31-s + (3 + 5.19i)37-s − 2·41-s + 8·43-s + (2 + 3.46i)47-s + (−5 + 8.66i)53-s + (−3 + 5.19i)59-s + (2 + 3.46i)61-s + (6 − 10.3i)67-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + (0.242 − 0.420i)17-s + (−0.229 − 0.397i)19-s + (0.834 + 1.44i)23-s + (−1.09 + 1.90i)25-s − 0.371·29-s + (0.359 − 0.622i)31-s + (0.493 + 0.854i)37-s − 0.312·41-s + 1.21·43-s + (0.291 + 0.505i)47-s + (−0.686 + 1.18i)53-s + (−0.390 + 0.676i)59-s + (0.256 + 0.443i)61-s + (0.733 − 1.26i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068747313\)
\(L(\frac12)\) \(\approx\) \(2.068747313\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.012219259332910874284935748867, −7.76679233843006839944444490973, −7.28804832595512287606106755915, −6.54286695062179918684502916235, −5.90947733141517000487513106681, −5.19524970094630884240125436941, −4.02964585873340844900127459211, −3.01999728223420432273379016585, −2.54388318054083608658843498475, −1.37169318387744329784635189955, 0.62749144485969038779114916663, 1.61354691657286283331511628625, 2.51413452533614293111305555332, 3.83627330332277467534591024871, 4.67870516459190559112360688322, 5.27013653752225416676142749432, 6.00979091530238650665108300516, 6.71030760177273511551002275405, 7.84031861569573747402608165382, 8.499494549783640319680022725911

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