↑

Properties

Label 2-3240-9.4-c1-0-25
Degree $2$
Conductor $3240$
Sign $0.173 + 0.984i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
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  + (−0.5 + 0.866i)5-s + (−2.13 − 3.70i)7-s + (0.637 + 1.10i)11-s + (−3.13 + 5.43i)13-s + 2·17-s + 19-s + (−0.137 + 0.238i)23-s + (−0.499 − 0.866i)25-s + (0.637 + 1.10i)29-s + (0.637 − 1.10i)31-s + 4.27·35-s + 4.54·37-s + (3.77 − 6.53i)41-s + (−2 − 3.46i)43-s + (−3.13 − 5.43i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.807 − 1.39i)7-s + (0.192 + 0.332i)11-s + (−0.870 + 1.50i)13-s + 0.485·17-s + 0.229·19-s + (−0.0286 + 0.0496i)23-s + (−0.0999 − 0.173i)25-s + (0.118 + 0.205i)29-s + (0.114 − 0.198i)31-s + 0.722·35-s + 0.747·37-s + (0.589 − 1.02i)41-s + (−0.304 − 0.528i)43-s + (−0.457 − 0.792i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.074841923\)
\(L(\frac12)\) \(\approx\) \(1.074841923\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (2.13 + 3.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.637 - 1.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.13 - 5.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (0.137 - 0.238i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.637 - 1.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.637 + 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + (-3.77 + 6.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.13 + 5.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.27T + 53T^{2} \)
59 \( 1 + (-6.5 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.27 - 5.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.27 + 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.725T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + (-2.27 - 3.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.27 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.82T + 89T^{2} \)
97 \( 1 + (8 + 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.455807225540462834214002409533, −7.42978968680404756991616881469, −7.03836172448920726817286488711, −6.56232871458853673207573016180, −5.43590690678611229897868269154, −4.31591032022742999178370831275, −3.93400762030274299023364035412, −2.93731378285775187473723937460, −1.79057216632657186770698716915, −0.39213207595486492653136790845, 0.988441127705432430257004561055, 2.63174143404559382185289653467, 2.96663573115802318698406795440, 4.13549531751586077961136025781, 5.28641907115237213210037992604, 5.63661864379034776542001094716, 6.43656546525843915890479731140, 7.44465769534680243588252108954, 8.199552401943200579518984165449, 8.716584123807434904046316709762

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