↑

Properties

Label 2-360-40.29-c1-0-11
Degree $2$
Conductor $360$
Sign $0.632 - 0.774i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + 1.41·2-s + 2.00·4-s + (−1.41 + 1.73i)5-s + 4.89i·7-s + 2.82·8-s + (−2.00 + 2.44i)10-s − 3.46i·11-s + 6.92i·14-s + 4.00·16-s + (−2.82 + 3.46i)20-s − 4.89i·22-s + (−0.999 − 4.89i)25-s + 9.79i·28-s − 10.3i·29-s + 10·31-s + 5.65·32-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + (−0.632 + 0.774i)5-s + 1.85i·7-s + 1.00·8-s + (−0.632 + 0.774i)10-s − 1.04i·11-s + 1.85i·14-s + 1.00·16-s + (−0.632 + 0.774i)20-s − 1.04i·22-s + (−0.199 − 0.979i)25-s + 1.85i·28-s − 1.92i·29-s + 1.79·31-s + 1.00·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03590 + 0.966031i\)
\(L(\frac12)\) \(\approx\) \(2.03590 + 0.966031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (1.41 - 1.73i)T \)
good7 \( 1 - 4.89iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19.5iT - 97T^{2} \)
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

−11.72714754467083221505180811711, −11.10791017253203975124783134897, −9.921335370243433831393799639094, −8.548185937603075931566784450343, −7.78643897706455061667956405243, −6.31954583954699879361530998778, −5.92099877812555013315550642710, −4.59890365192795359426966641373, −3.21382196039481501191440307182, −2.44764899536311490695836001089, 1.33458986035344808092351792073, 3.40291388318671117170332009376, 4.40062229500986507554006507399, 4.94000032488637973811100299843, 6.64361071551697573271240010768, 7.34110727449173129191484639931, 8.152683965029034971226494637639, 9.746216393147639093146409367575, 10.60589177834492415788662520152, 11.42439615359571805888595087366

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