↑

Properties

Label 2-1380-92.91-c1-0-27
Degree $2$
Conductor $1380$
Sign $-0.993 - 0.109i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (1.26 + 0.625i)2-s + i·3-s + (1.21 + 1.58i)4-s + i·5-s + (−0.625 + 1.26i)6-s − 1.40·7-s + (0.551 + 2.77i)8-s − 9-s + (−0.625 + 1.26i)10-s − 3.17·11-s + (−1.58 + 1.21i)12-s + 1.35·13-s + (−1.78 − 0.878i)14-s − 15-s + (−1.03 + 3.86i)16-s + 2.06i·17-s + ⋯
L(s)  = 1  + (0.896 + 0.442i)2-s + 0.577i·3-s + (0.608 + 0.793i)4-s + 0.447i·5-s + (−0.255 + 0.517i)6-s − 0.531·7-s + (0.195 + 0.980i)8-s − 0.333·9-s + (−0.197 + 0.401i)10-s − 0.958·11-s + (−0.458 + 0.351i)12-s + 0.375·13-s + (−0.476 − 0.234i)14-s − 0.258·15-s + (−0.258 + 0.965i)16-s + 0.499i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.993 - 0.109i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.993 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.995362196\)
\(L(\frac12)\) \(\approx\) \(1.995362196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.625i)T \)
3 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + (3.32 - 3.46i)T \)
good7 \( 1 + 1.40T + 7T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 - 2.06iT - 17T^{2} \)
19 \( 1 + 1.06T + 19T^{2} \)
29 \( 1 + 1.11T + 29T^{2} \)
31 \( 1 - 1.91iT - 31T^{2} \)
37 \( 1 + 0.487iT - 37T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 5.44T + 43T^{2} \)
47 \( 1 - 2.44iT - 47T^{2} \)
53 \( 1 - 1.02iT - 53T^{2} \)
59 \( 1 - 0.773iT - 59T^{2} \)
61 \( 1 + 0.621iT - 61T^{2} \)
67 \( 1 - 3.87T + 67T^{2} \)
71 \( 1 + 4.43iT - 71T^{2} \)
73 \( 1 - 8.92T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 5.46T + 83T^{2} \)
89 \( 1 - 6.20iT - 89T^{2} \)
97 \( 1 + 4.97iT - 97T^{2} \)
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

−10.11772218789383890876171191334, −9.136052815800005451281227423813, −8.126658366481581754008840106715, −7.50573506329033290807867081524, −6.43687669716375574232773121447, −5.84078588039892096025644045364, −4.97383952442497607596913987670, −3.94121988639059123259861584126, −3.24953219881085095266112105223, −2.23788275344140764350606499108, 0.55636303881416288019156788995, 2.02943292305210390911813635722, 2.90857352894276719277111578689, 3.99685206645869068847448445341, 4.98294105796490426033204074258, 5.81325474845759374991112060225, 6.51074005658853908746883966093, 7.44199774765170356494214387026, 8.288497769341054296622285974991, 9.344736786466034465598504524357

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