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Properties

Label 2-1980-11.4-c1-0-1
Degree $2$
Conductor $1980$
Sign $-0.739 - 0.673i$
Analytic cond. $15.8103$
Root an. cond. $3.97622$
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  + (−0.809 + 0.587i)5-s + (−1.26 − 3.88i)7-s + (2.91 − 1.57i)11-s + (−5.72 − 4.15i)13-s + (0.657 − 0.477i)17-s + (−2.56 + 7.90i)19-s − 7.02·23-s + (0.309 − 0.951i)25-s + (2.19 + 6.75i)29-s + (3.69 + 2.68i)31-s + (3.30 + 2.40i)35-s + (−0.971 − 2.98i)37-s + (−1.96 + 6.05i)41-s + 1.35·43-s + (−0.858 + 2.64i)47-s + ⋯
L(s)  = 1  + (−0.361 + 0.262i)5-s + (−0.477 − 1.46i)7-s + (0.880 − 0.474i)11-s + (−1.58 − 1.15i)13-s + (0.159 − 0.115i)17-s + (−0.588 + 1.81i)19-s − 1.46·23-s + (0.0618 − 0.190i)25-s + (0.407 + 1.25i)29-s + (0.663 + 0.482i)31-s + (0.558 + 0.405i)35-s + (−0.159 − 0.491i)37-s + (−0.307 + 0.946i)41-s + 0.206·43-s + (−0.125 + 0.385i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.739 - 0.673i$
Analytic conductor: \(15.8103\)
Root analytic conductor: \(3.97622\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1980,\ (\ :1/2),\ -0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1287900768\)
\(L(\frac12)\) \(\approx\) \(0.1287900768\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.91 + 1.57i)T \)
good7 \( 1 + (1.26 + 3.88i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.72 + 4.15i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.657 + 0.477i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.56 - 7.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.02T + 23T^{2} \)
29 \( 1 + (-2.19 - 6.75i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.69 - 2.68i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.971 + 2.98i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.96 - 6.05i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.35T + 43T^{2} \)
47 \( 1 + (0.858 - 2.64i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.62 - 4.08i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.34 - 4.12i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.12 + 5.90i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + (7.22 - 5.24i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.649 - 1.99i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.1 + 7.38i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.3 - 8.21i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.800T + 89T^{2} \)
97 \( 1 + (3.90 + 2.83i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.815673059705510720935205841432, −8.497440081355589834959183700599, −7.81776256077956096994304270373, −7.19122061813701083586623432535, −6.43470359384137631472929698868, −5.54025563537647664993342925948, −4.33953443711286487759470240696, −3.74001213797885012220629025380, −2.86612916062127367389833051668, −1.28646645614749908221299166502, 0.04681583418460220292706391386, 2.08050131254427150981638929134, 2.61538522672079041266212167956, 4.11197745023772250657486716019, 4.68165429885818806907717276965, 5.69192948815267135211417384013, 6.59663930630293380332585256597, 7.13548749500909426065838134116, 8.285098359487690797541175901304, 8.932992121308093096134707923312

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