↑

Properties

Label 2-3216-201.155-c0-0-0
Degree $2$
Conductor $3216$
Sign $-0.00408 - 0.999i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 + 0.909i)3-s + (−0.0930 + 0.268i)7-s + (−0.654 − 0.755i)9-s + (0.581 − 0.299i)13-s + (0.271 + 0.785i)19-s + (−0.205 − 0.196i)21-s + (0.841 + 0.540i)25-s + (0.959 − 0.281i)27-s + (0.419 + 0.216i)31-s + (−0.841 + 1.45i)37-s + (0.0311 + 0.653i)39-s + (0.0135 + 0.0941i)43-s + (0.722 + 0.568i)49-s + (−0.827 − 0.0789i)57-s + (−0.469 + 1.93i)61-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)3-s + (−0.0930 + 0.268i)7-s + (−0.654 − 0.755i)9-s + (0.581 − 0.299i)13-s + (0.271 + 0.785i)19-s + (−0.205 − 0.196i)21-s + (0.841 + 0.540i)25-s + (0.959 − 0.281i)27-s + (0.419 + 0.216i)31-s + (−0.841 + 1.45i)37-s + (0.0311 + 0.653i)39-s + (0.0135 + 0.0941i)43-s + (0.722 + 0.568i)49-s + (−0.827 − 0.0789i)57-s + (−0.469 + 1.93i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00408 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00408 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3216\)    =    \(2^{4} \cdot 3 \cdot 67\)
Sign: $-0.00408 - 0.999i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3216} (1361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3216,\ (\ :0),\ -0.00408 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016154248\)
\(L(\frac12)\) \(\approx\) \(1.016154248\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
good5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.0930 - 0.268i)T + (-0.786 - 0.618i)T^{2} \)
11 \( 1 + (0.888 - 0.458i)T^{2} \)
13 \( 1 + (-0.581 + 0.299i)T + (0.580 - 0.814i)T^{2} \)
17 \( 1 + (-0.723 - 0.690i)T^{2} \)
19 \( 1 + (-0.271 - 0.785i)T + (-0.786 + 0.618i)T^{2} \)
23 \( 1 + (0.327 - 0.945i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.419 - 0.216i)T + (0.580 + 0.814i)T^{2} \)
37 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.235 + 0.971i)T^{2} \)
43 \( 1 + (-0.0135 - 0.0941i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.981 + 0.189i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.469 - 1.93i)T + (-0.888 - 0.458i)T^{2} \)
71 \( 1 + (-0.723 + 0.690i)T^{2} \)
73 \( 1 + (-0.341 + 1.40i)T + (-0.888 - 0.458i)T^{2} \)
79 \( 1 + (-0.0845 + 1.77i)T + (-0.995 - 0.0950i)T^{2} \)
83 \( 1 + (-0.0475 + 0.998i)T^{2} \)
89 \( 1 + (0.654 - 0.755i)T^{2} \)
97 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.972155829967854925094469877611, −8.545571617611115804992337962534, −7.57912449019460121398742675980, −6.58450909550910286657654794052, −5.90961516661610373585798455714, −5.22440560916524412373739109154, −4.44153113275145054264768215713, −3.52243017779081055700982619735, −2.84827737601748570911327684617, −1.27465537188293684278049773437, 0.73272170749903416030883839501, 1.91773980057248997901174045934, 2.88042953014284556542827537287, 3.96975215347357489192414981407, 4.97413706559138119852415789647, 5.68885074431202606419030143436, 6.62211084409252700045184531432, 6.97144505813351156655876870120, 7.85576006459879458081689869347, 8.553360181232685985398690195054

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