↑

Properties

Label 2-3528-7.4-c1-0-15
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (−3 + 5.19i)11-s + 6·13-s + (−1 + 1.73i)17-s + (2 + 3.46i)19-s + (−1 − 1.73i)23-s + (0.500 − 0.866i)25-s + 8·29-s + (2 − 3.46i)31-s + (3 + 5.19i)37-s − 10·41-s − 4·43-s + (−2 − 3.46i)47-s + (2 − 3.46i)53-s − 12·55-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (−0.904 + 1.56i)11-s + 1.66·13-s + (−0.242 + 0.420i)17-s + (0.458 + 0.794i)19-s + (−0.208 − 0.361i)23-s + (0.100 − 0.173i)25-s + 1.48·29-s + (0.359 − 0.622i)31-s + (0.493 + 0.854i)37-s − 1.56·41-s − 0.609·43-s + (−0.291 − 0.505i)47-s + (0.274 − 0.475i)53-s − 1.61·55-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

\(L(1)\) \(\approx\) \(1.895436648\)
\(L(\frac12)\) \(\approx\) \(1.895436648\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
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 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.548731714191857797173996603179, −8.181642734127203393826297818771, −7.20168697974625027682281684794, −6.51724188660796998538176074111, −5.97940996027913103930539336202, −4.97156680894205134324518776623, −4.18001771576302534662991864852, −3.18502558688775385687432950433, −2.33101633129175884736997843814, −1.38605028238985385935584744063, 0.59428116124991897678182422831, 1.48660389731909005918055042225, 2.90088395096359887441530857054, 3.47804354098562059591615173365, 4.72620104924441527983848819359, 5.32127685999385607990761362779, 6.05868621743402411021196542835, 6.68450368100258285853207599550, 7.86569791232421248871365970178, 8.489205427224337275673642677005

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