↑

Properties

Label 2-1120-35.34-c0-0-2
Degree $2$
Conductor $1120$
Sign $0.707 - 0.707i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
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  + 1.41·3-s + i·5-s + (−0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 + 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s − 1.00i·49-s + ⋯
L(s)  = 1  + 1.41·3-s + i·5-s + (−0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 + 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (−0.707 − 0.707i)35-s + 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504609325\)
\(L(\frac12)\) \(\approx\) \(1.504609325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
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

−9.951072030128261790860211124697, −9.322581398509889351506150269757, −8.414677168539626117200315587999, −7.930783705846155493843864390680, −6.69537744581888639462228010301, −6.31935010476225589123642825744, −4.81691452895326005921719859864, −3.54039684525809606420512015719, −2.89709334595176181591152831215, −2.18861563401523290385669901959, 1.35849802328338712209255188712, 2.74726002137464069677077475211, 3.66937449300907883328296566916, 4.45278513780856770361299141264, 5.62460238959048229618598604484, 6.81632344585498536511998192800, 7.68896527229820189225773134837, 8.351414923921519626933471680704, 9.094539555409779792965332100456, 9.693164450475603469305425520211

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