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Properties

Label 2-3528-7.2-c1-0-13
Degree $2$
Conductor $3528$
Sign $-0.827 - 0.561i$
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  + (−1.70 + 2.95i)5-s + (2.41 + 4.18i)11-s − 1.41·13-s + (3.12 + 5.40i)17-s + (0.585 − 1.01i)19-s + (−0.414 + 0.717i)23-s + (−3.32 − 5.76i)25-s + 8.48·29-s + (5.41 + 9.37i)31-s + (4.82 − 8.36i)37-s − 3.41·41-s − 8·43-s + (−0.585 + 1.01i)47-s + (4.65 + 8.06i)53-s − 16.4·55-s + ⋯
L(s)  = 1  + (−0.763 + 1.32i)5-s + (0.727 + 1.26i)11-s − 0.392·13-s + (0.757 + 1.31i)17-s + (0.134 − 0.232i)19-s + (−0.0863 + 0.149i)23-s + (−0.665 − 1.15i)25-s + 1.57·29-s + (0.972 + 1.68i)31-s + (0.793 − 1.37i)37-s − 0.533·41-s − 1.21·43-s + (−0.0854 + 0.147i)47-s + (0.639 + 1.10i)53-s − 2.22·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427557037\)
\(L(\frac12)\) \(\approx\) \(1.427557037\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.41 - 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.585 + 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.414 - 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + (-5.41 - 9.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.82 + 8.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (0.585 - 1.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.65 - 8.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.94 - 5.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 + (-1.53 - 2.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.82 + 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (7.36 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.756943351503415748650596822420, −7.995531370832928895806225181123, −7.29727516428173381908257217454, −6.75459539706562853477101896323, −6.15496192979023363137688213609, −4.92764597896222208164230206560, −4.15751542226248596927322492767, −3.38653103590478847443411964380, −2.58129074279358137913470463237, −1.40839519291490197859445936289, 0.50413212179843701597384296417, 1.17624977946116545891388711456, 2.77758630557693260001539365491, 3.61322556896815594836589960680, 4.55906016047068944470063526822, 5.03622018234659522068019053655, 5.99410248554220954213580216406, 6.75549499810040010782391747853, 7.86675275925895924484030412766, 8.206941051460554830878640501782

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