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Properties

Label 2-1980-11.3-c1-0-7
Degree $2$
Conductor $1980$
Sign $0.578 - 0.815i$
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 + (−0.130 + 0.402i)7-s + (3.18 + 0.911i)11-s + (−2.03 + 1.47i)13-s + (2.31 + 1.68i)17-s + (−0.447 − 1.37i)19-s + 3.37·23-s + (0.309 + 0.951i)25-s + (1.26 − 3.88i)29-s + (−1.67 + 1.21i)31-s + (−0.342 + 0.248i)35-s + (−2.01 + 6.21i)37-s + (1.70 + 5.25i)41-s − 8.45·43-s + (2.68 + 8.25i)47-s + ⋯
L(s)  = 1  + (0.361 + 0.262i)5-s + (−0.0494 + 0.152i)7-s + (0.961 + 0.274i)11-s + (−0.563 + 0.409i)13-s + (0.562 + 0.408i)17-s + (−0.102 − 0.316i)19-s + 0.704·23-s + (0.0618 + 0.190i)25-s + (0.234 − 0.721i)29-s + (−0.300 + 0.218i)31-s + (−0.0578 + 0.0420i)35-s + (−0.331 + 1.02i)37-s + (0.266 + 0.820i)41-s − 1.28·43-s + (0.391 + 1.20i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(15.8103\)
Root analytic conductor: \(3.97622\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1980,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.876421040\)
\(L(\frac12)\) \(\approx\) \(1.876421040\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.18 - 0.911i)T \)
good7 \( 1 + (0.130 - 0.402i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.03 - 1.47i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.31 - 1.68i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.447 + 1.37i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + (-1.26 + 3.88i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.67 - 1.21i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.01 - 6.21i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.70 - 5.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.45T + 43T^{2} \)
47 \( 1 + (-2.68 - 8.25i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.27 + 4.56i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.00136 + 0.00420i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.24 + 2.35i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.67T + 67T^{2} \)
71 \( 1 + (-2.00 - 1.45i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.730 - 2.24i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.8 + 8.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.0 - 8.04i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (8.40 - 6.10i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.397002165016268210191727764979, −8.596377574492459097600817366438, −7.68469307377337324155507678420, −6.79467364368409445335387059269, −6.29379160795935048003895087050, −5.25226244716352824126064782212, −4.41961266323137324858775648633, −3.40927714936942056668068091742, −2.38041336114778165132237620075, −1.25843053551625831279331009362, 0.75686140049624829553512162246, 1.98602073331175623499726065726, 3.20066145278748336411720522247, 4.05332004662505473677101163426, 5.16543434681784584927196184300, 5.72244394051930555662262134618, 6.81939685113880101113305806798, 7.34012801832050741068134304954, 8.433328854410352714661013579188, 9.036679448420455059032252697765

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