Properties

Label 2-1380-92.91-c1-0-95
Degree $2$
Conductor $1380$
Sign $-0.870 - 0.492i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.382i)2-s i·3-s + (1.70 − 1.04i)4-s + i·5-s + (−0.382 − 1.36i)6-s − 3.81·7-s + (1.92 − 2.07i)8-s − 9-s + (0.382 + 1.36i)10-s − 3.29·11-s + (−1.04 − 1.70i)12-s − 6.74·13-s + (−5.18 + 1.45i)14-s + 15-s + (1.83 − 3.55i)16-s + 2.05i·17-s + ⋯
L(s)  = 1  + (0.962 − 0.270i)2-s − 0.577i·3-s + (0.853 − 0.520i)4-s + 0.447i·5-s + (−0.156 − 0.555i)6-s − 1.44·7-s + (0.681 − 0.732i)8-s − 0.333·9-s + (0.120 + 0.430i)10-s − 0.992·11-s + (−0.300 − 0.492i)12-s − 1.87·13-s + (−1.38 + 0.389i)14-s + 0.258·15-s + (0.457 − 0.889i)16-s + 0.498i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.870 - 0.492i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3513982613\)
\(L(\frac12)\) \(\approx\) \(0.3513982613\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.382i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (-2.33 - 4.19i)T \)
good7 \( 1 + 3.81T + 7T^{2} \)
11 \( 1 + 3.29T + 11T^{2} \)
13 \( 1 + 6.74T + 13T^{2} \)
17 \( 1 - 2.05iT - 17T^{2} \)
19 \( 1 + 4.83T + 19T^{2} \)
29 \( 1 - 4.01T + 29T^{2} \)
31 \( 1 - 0.0146iT - 31T^{2} \)
37 \( 1 + 6.85iT - 37T^{2} \)
41 \( 1 + 4.62T + 41T^{2} \)
43 \( 1 - 4.10T + 43T^{2} \)
47 \( 1 + 4.94iT - 47T^{2} \)
53 \( 1 + 8.07iT - 53T^{2} \)
59 \( 1 + 7.38iT - 59T^{2} \)
61 \( 1 + 2.85iT - 61T^{2} \)
67 \( 1 - 0.919T + 67T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 9.94T + 83T^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 + 4.51iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.394987458892717211170245934585, −8.006236302969281137962402786420, −7.10921000429704731144447868882, −6.69766168847697379644498203166, −5.76895937933245473250711241789, −4.96983347868702347465239118680, −3.73204074440811854628334253046, −2.79650140502329232694676042442, −2.18850952949576554835016320583, −0.088818735096042755683040937562, 2.57631538119459847364143684375, 2.97535718851918848194670098352, 4.38365636945574369164932773629, 4.84521932703430707355349769765, 5.79276817716955227286869948864, 6.65939141731969829197123082027, 7.40191551532354296065160448034, 8.381615564400437867994774884089, 9.334222288256155686231485106123, 10.18526948685065638467421759372

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