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Properties

Label 2-1380-92.91-c1-0-94
Degree $2$
Conductor $1380$
Sign $-0.839 - 0.543i$
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  + (1.21 − 0.729i)2-s + i·3-s + (0.936 − 1.76i)4-s − i·5-s + (0.729 + 1.21i)6-s − 3.55·7-s + (−0.154 − 2.82i)8-s − 9-s + (−0.729 − 1.21i)10-s − 4.26·11-s + (1.76 + 0.936i)12-s − 0.943·13-s + (−4.30 + 2.59i)14-s + 15-s + (−2.24 − 3.30i)16-s + 4.88i·17-s + ⋯
L(s)  = 1  + (0.856 − 0.515i)2-s + 0.577i·3-s + (0.468 − 0.883i)4-s − 0.447i·5-s + (0.297 + 0.494i)6-s − 1.34·7-s + (−0.0544 − 0.998i)8-s − 0.333·9-s + (−0.230 − 0.383i)10-s − 1.28·11-s + (0.510 + 0.270i)12-s − 0.261·13-s + (−1.15 + 0.692i)14-s + 0.258·15-s + (−0.561 − 0.827i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.839 - 0.543i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06912305081\)
\(L(\frac12)\) \(\approx\) \(0.06912305081\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 + 0.729i)T \)
3 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (0.416 - 4.77i)T \)
good7 \( 1 + 3.55T + 7T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 + 0.943T + 13T^{2} \)
17 \( 1 - 4.88iT - 17T^{2} \)
19 \( 1 - 0.383T + 19T^{2} \)
29 \( 1 + 7.10T + 29T^{2} \)
31 \( 1 - 5.11iT - 31T^{2} \)
37 \( 1 + 0.0835iT - 37T^{2} \)
41 \( 1 - 8.97T + 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 + 2.89iT - 47T^{2} \)
53 \( 1 + 3.78iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 7.09iT - 61T^{2} \)
67 \( 1 - 9.32T + 67T^{2} \)
71 \( 1 + 9.71iT - 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 + 8.34T + 79T^{2} \)
83 \( 1 - 1.59T + 83T^{2} \)
89 \( 1 + 5.90iT - 89T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−9.564447467165944538690707737912, −8.421474682158127734346768508945, −7.33827469215978928780417096648, −6.30458236157166088328290220066, −5.55745898681998543458053573124, −4.88650160348422492899857454289, −3.70842872676028286066251642052, −3.19925231175021700057277074739, −1.94782329515727081607647060373, −0.01859551588468466599349315960, 2.52263559556286124462683998171, 2.89328146535730281401339541708, 4.07733734343431129385484087025, 5.28206289460916306531303112486, 5.96146903677023358200979507638, 6.81765288885388545812402621446, 7.35565164920131993465638118818, 8.082358138000234991458936831903, 9.195826062331346577064698462501, 10.05968390924815031807093771096

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