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Properties

Label 2-1380-92.91-c1-0-7
Degree $2$
Conductor $1380$
Sign $0.837 - 0.546i$
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  + (0.0496 − 1.41i)2-s − i·3-s + (−1.99 − 0.140i)4-s − i·5-s + (−1.41 − 0.0496i)6-s − 2.88·7-s + (−0.297 + 2.81i)8-s − 9-s + (−1.41 − 0.0496i)10-s − 1.07·11-s + (−0.140 + 1.99i)12-s − 2.06·13-s + (−0.143 + 4.07i)14-s − 15-s + (3.96 + 0.560i)16-s − 6.90i·17-s + ⋯
L(s)  = 1  + (0.0351 − 0.999i)2-s − 0.577i·3-s + (−0.997 − 0.0702i)4-s − 0.447i·5-s + (−0.576 − 0.0202i)6-s − 1.08·7-s + (−0.105 + 0.994i)8-s − 0.333·9-s + (−0.446 − 0.0157i)10-s − 0.324·11-s + (−0.0405 + 0.575i)12-s − 0.573·13-s + (−0.0382 + 1.08i)14-s − 0.258·15-s + (0.990 + 0.140i)16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.837 - 0.546i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2185411371\)
\(L(\frac12)\) \(\approx\) \(0.2185411371\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0496 + 1.41i)T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
23 \( 1 + (3.82 - 2.89i)T \)
good7 \( 1 + 2.88T + 7T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 + 2.06T + 13T^{2} \)
17 \( 1 + 6.90iT - 17T^{2} \)
19 \( 1 - 0.565T + 19T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
37 \( 1 - 6.37iT - 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 0.639T + 43T^{2} \)
47 \( 1 - 8.82iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 + 6.13iT - 59T^{2} \)
61 \( 1 + 9.75iT - 61T^{2} \)
67 \( 1 + 4.68T + 67T^{2} \)
71 \( 1 + 3.80iT - 71T^{2} \)
73 \( 1 + 4.92T + 73T^{2} \)
79 \( 1 - 3.27T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 9.63iT - 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.556232476910584747095702346185, −9.214505014564021205708232352148, −8.131783384325205433599580587188, −7.34020167955295736706668067480, −6.31326306184828576823628803456, −5.30080721914028176929784630766, −4.53725534376534682454487962914, −3.19637691222942863994911568253, −2.62345547815216588320955618625, −1.18451512984227671344132584110, 0.096669359985089481949797492595, 2.54710881340236454466857309572, 3.76758226669822476364120888870, 4.30375157951172569495059458764, 5.68296200528291720430172816328, 6.08499087216669713914750333968, 6.98612143147232551261568394694, 7.86352762862001920079952380914, 8.616335097821659042427040849477, 9.560439318067007374523761388514

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