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Properties

Label 2-1380-92.91-c1-0-78
Degree $2$
Conductor $1380$
Sign $-0.999 - 0.000974i$
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  + (−1.39 − 0.257i)2-s − i·3-s + (1.86 + 0.717i)4-s + i·5-s + (−0.257 + 1.39i)6-s − 2.59·7-s + (−2.41 − 1.47i)8-s − 9-s + (0.257 − 1.39i)10-s + 1.75·11-s + (0.717 − 1.86i)12-s − 2.03·13-s + (3.61 + 0.670i)14-s + 15-s + (2.97 + 2.67i)16-s − 3.95i·17-s + ⋯
L(s)  = 1  + (−0.983 − 0.182i)2-s − 0.577i·3-s + (0.933 + 0.358i)4-s + 0.447i·5-s + (−0.105 + 0.567i)6-s − 0.982·7-s + (−0.852 − 0.522i)8-s − 0.333·9-s + (0.0815 − 0.439i)10-s + 0.528·11-s + (0.207 − 0.538i)12-s − 0.564·13-s + (0.965 + 0.179i)14-s + 0.258·15-s + (0.742 + 0.669i)16-s − 0.959i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.000974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.999 - 0.000974i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.999 - 0.000974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2308935514\)
\(L(\frac12)\) \(\approx\) \(0.2308935514\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.257i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (-4.47 + 1.71i)T \)
good7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 - 1.75T + 11T^{2} \)
13 \( 1 + 2.03T + 13T^{2} \)
17 \( 1 + 3.95iT - 17T^{2} \)
19 \( 1 - 5.10T + 19T^{2} \)
29 \( 1 + 7.32T + 29T^{2} \)
31 \( 1 - 4.91iT - 31T^{2} \)
37 \( 1 - 0.890iT - 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 + 2.70iT - 47T^{2} \)
53 \( 1 + 2.16iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 7.39iT - 61T^{2} \)
67 \( 1 + 8.20T + 67T^{2} \)
71 \( 1 + 7.48iT - 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 + 9.80T + 83T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 - 0.186iT - 97T^{2} \)
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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

−9.306115244017411081430998804834, −8.474051703823363747190856419562, −7.28235880409716096044244409673, −7.08197956979513888097998687944, −6.27401852770338036309160812584, −5.17387406976178949456018974940, −3.39782945806444804713740373337, −2.89594715564875515447061793344, −1.58068801637653496050179120694, −0.13065446489841014820952220426, 1.47109708826024084498195667965, 2.92798162581173415531371561148, 3.82141380836968405923709025902, 5.19201984854263827156067695514, 5.93521374844825474760287616397, 6.85471190600115622376255353755, 7.63341080196482471605054586380, 8.594553067846265233620026933599, 9.363428715242161526023131245621, 9.695452877745648345185951102560

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