↑

Properties

Label 2-3584-3584.2085-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.824 - 0.565i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
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  + (0.671 + 0.740i)2-s + (−0.0980 + 0.995i)4-s + (−0.146 − 0.989i)7-s + (−0.803 + 0.595i)8-s + (0.514 − 0.857i)9-s + (0.0153 − 0.0466i)11-s + (0.634 − 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (0.0448 − 0.0198i)22-s + (0.326 − 0.360i)23-s + (0.336 + 0.941i)25-s + (0.998 − 0.0490i)28-s + (1.96 + 0.144i)29-s + (−0.514 − 0.857i)32-s + ⋯
L(s)  = 1  + (0.671 + 0.740i)2-s + (−0.0980 + 0.995i)4-s + (−0.146 − 0.989i)7-s + (−0.803 + 0.595i)8-s + (0.514 − 0.857i)9-s + (0.0153 − 0.0466i)11-s + (0.634 − 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (0.0448 − 0.0198i)22-s + (0.326 − 0.360i)23-s + (0.336 + 0.941i)25-s + (0.998 − 0.0490i)28-s + (1.96 + 0.144i)29-s + (−0.514 − 0.857i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2085, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.839487646\)
\(L(\frac12)\) \(\approx\) \(1.839487646\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.671 - 0.740i)T \)
7 \( 1 + (0.146 + 0.989i)T \)
good3 \( 1 + (-0.514 + 0.857i)T^{2} \)
5 \( 1 + (-0.336 - 0.941i)T^{2} \)
11 \( 1 + (-0.0153 + 0.0466i)T + (-0.803 - 0.595i)T^{2} \)
13 \( 1 + (0.427 + 0.903i)T^{2} \)
17 \( 1 + (0.980 - 0.195i)T^{2} \)
19 \( 1 + (0.998 - 0.0490i)T^{2} \)
23 \( 1 + (-0.326 + 0.360i)T + (-0.0980 - 0.995i)T^{2} \)
29 \( 1 + (-1.96 - 0.144i)T + (0.989 + 0.146i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.585 + 0.225i)T + (0.740 - 0.671i)T^{2} \)
41 \( 1 + (0.773 + 0.634i)T^{2} \)
43 \( 1 + (-1.45 + 0.403i)T + (0.857 - 0.514i)T^{2} \)
47 \( 1 + (0.831 + 0.555i)T^{2} \)
53 \( 1 + (-0.244 + 0.0180i)T + (0.989 - 0.146i)T^{2} \)
59 \( 1 + (-0.427 + 0.903i)T^{2} \)
61 \( 1 + (-0.242 - 0.970i)T^{2} \)
67 \( 1 + (-0.0652 - 0.529i)T + (-0.970 + 0.242i)T^{2} \)
71 \( 1 + (-0.229 + 0.914i)T + (-0.881 - 0.471i)T^{2} \)
73 \( 1 + (0.956 + 0.290i)T^{2} \)
79 \( 1 + (1.59 + 0.852i)T + (0.555 + 0.831i)T^{2} \)
83 \( 1 + (-0.740 - 0.671i)T^{2} \)
89 \( 1 + (0.0980 - 0.995i)T^{2} \)
97 \( 1 + (0.382 - 0.923i)T^{2} \)
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

−8.725753152646468640716660858843, −7.80791678783340249153498303072, −7.16382360664405487742630395524, −6.63394711326636147547259061521, −5.96496036582345843880693522730, −4.91040177336610898492664912486, −4.26252467991949690013935338400, −3.56742879540292981985154188598, −2.72064053943493505687597323328, −1.03471089610191301461752851008, 1.26873811405448189861541511631, 2.43783243244713883449192797527, 2.85970475046393605644915370275, 4.15277085898876820160131450466, 4.75577545219503652458472289636, 5.50222834003325611571265214555, 6.24686904427773216025084707036, 6.98373379364895191613883799211, 8.082931919532862680503121729027, 8.732117642487740214916467730628

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