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Properties

Label 2-1320-11.3-c1-0-17
Degree $2$
Conductor $1320$
Sign $0.199 + 0.979i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (−0.596 + 1.83i)7-s + (−0.809 + 0.587i)9-s + (−2.49 − 2.18i)11-s + (0.941 − 0.683i)13-s + (0.309 − 0.951i)15-s + (0.674 + 0.489i)17-s + (−2.37 − 7.31i)19-s + 1.92·21-s + 2.39·23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (1.87 − 5.77i)29-s + (2.80 − 2.03i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.361 + 0.262i)5-s + (−0.225 + 0.693i)7-s + (−0.269 + 0.195i)9-s + (−0.753 − 0.657i)11-s + (0.261 − 0.189i)13-s + (0.0797 − 0.245i)15-s + (0.163 + 0.118i)17-s + (−0.545 − 1.67i)19-s + 0.420·21-s + 0.498·23-s + (0.0618 + 0.190i)25-s + (0.155 + 0.113i)27-s + (0.348 − 1.07i)29-s + (0.504 − 0.366i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.199 + 0.979i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.199 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339211421\)
\(L(\frac12)\) \(\approx\) \(1.339211421\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.49 + 2.18i)T \)
good7 \( 1 + (0.596 - 1.83i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.941 + 0.683i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.674 - 0.489i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.37 + 7.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 + (-1.87 + 5.77i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.80 + 2.03i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.90 + 8.92i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.18 - 6.72i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + (3.54 + 10.9i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.66 + 2.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.82 - 5.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.29 - 2.39i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.84T + 67T^{2} \)
71 \( 1 + (4.76 + 3.45i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.88 + 5.79i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.07 - 3.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.59 + 6.96i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.35T + 89T^{2} \)
97 \( 1 + (-0.915 + 0.665i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.307516694204080629362194886922, −8.686851480198500949027243640137, −7.81427919733159166769075527343, −6.94857410748276225961786710160, −6.03373931390007392019992062403, −5.56246188957449476182769304067, −4.39640633536113929322721022896, −2.89143865333904058722227613270, −2.35336023402606436489350422309, −0.61125116711552936096701728494, 1.29269112360991566253096591648, 2.74471684232234749694657889068, 3.89795485884014018706492267247, 4.67636306635042816254766081842, 5.57809489170143463140363876145, 6.43182732157001283546510313002, 7.39530803486983037113857559099, 8.229843911892502165948647676742, 9.120127119418242565815062438060, 10.00012492853571769100446929646

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