↑

Properties

Label 2-2940-7.4-c1-0-20
Degree $2$
Conductor $2940$
Sign $-0.386 + 0.922i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 4·13-s − 0.999·15-s + (−1 + 1.73i)17-s + (−1 − 1.73i)19-s + (−2 − 3.46i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + 6·29-s + (1 − 1.73i)31-s + (−0.999 − 1.73i)33-s + (−5 − 8.66i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 1.10·13-s − 0.258·15-s + (−0.242 + 0.420i)17-s + (−0.229 − 0.397i)19-s + (−0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (0.179 − 0.311i)31-s + (−0.174 − 0.301i)33-s + (−0.821 − 1.42i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.754155203\)
\(L(\frac12)\) \(\approx\) \(1.754155203\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
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 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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

−8.655939034338345697155057699080, −7.907874656403813234861817784412, −6.97809132272929596079752408086, −6.26956590935373123519379768342, −5.62487466143993170523326065425, −4.43729587426430170656510290499, −3.77010574966112410097961211953, −2.76406401030701506625382780557, −1.64937687243616779230932029827, −0.56477112115496538302130946808, 1.34330332175008026168851722396, 2.54854771601251225415914081737, 3.50575859436075359254156725618, 4.13878536680193821875103740551, 5.03653109322354569477218550042, 5.97367861583938014292527882200, 6.76327145842183402637119278082, 7.46768965765711649883025369133, 8.498390417704503515117102502873, 8.757794150409406497292650805055

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