↑

Properties

Label 2-960-240.29-c0-0-1
Degree $2$
Conductor $960$
Sign $0.923 - 0.382i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
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  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + 1.00i·9-s + 1.00·15-s − 1.41·17-s + (1 + i)19-s − 1.41i·23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)45-s − 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s + 1.41i·57-s + (1 + i)61-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + 1.00i·9-s + 1.00·15-s − 1.41·17-s + (1 + i)19-s − 1.41i·23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)45-s − 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s + 1.41i·57-s + (1 + i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375047986\)
\(L(\frac12)\) \(\approx\) \(1.375047986\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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

−10.05138144830170383170687185455, −9.496499585742355701194734615427, −8.635544803135460749451788695433, −8.160980986375860629738448735087, −6.89582988152375854335416825106, −5.82863166938249298973048772723, −4.87735690847406969783657677496, −4.16669900258668777363601989195, −2.88035360203597166748712738625, −1.79310339574997505454639337843, 1.65781942631712337322272859052, 2.67837960823390658538611852837, 3.54961597074933272583482833432, 5.00721713175100405242841594570, 6.14003037804188266670919153498, 6.88848049230921251143996625197, 7.49156758015824825761964801315, 8.562263352978267541676388236071, 9.386362882517186432702955875513, 9.894572229778655794612678331111

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