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Properties

Label 2-1320-5.4-c1-0-17
Degree $2$
Conductor $1320$
Sign $0.998 - 0.0566i$
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  − i·3-s + (2.23 − 0.126i)5-s + 3.92i·7-s − 9-s − 11-s − 2.60i·13-s + (−0.126 − 2.23i)15-s − 2.54i·17-s + 4.21·19-s + 3.92·21-s + 3.65i·23-s + (4.96 − 0.565i)25-s + i·27-s + 3.40·29-s + 9.61·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.998 − 0.0566i)5-s + 1.48i·7-s − 0.333·9-s − 0.301·11-s − 0.723i·13-s + (−0.0326 − 0.576i)15-s − 0.616i·17-s + 0.966·19-s + 0.856·21-s + 0.762i·23-s + (0.993 − 0.113i)25-s + 0.192i·27-s + 0.631·29-s + 1.72·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.998 - 0.0566i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.998 - 0.0566i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013824711\)
\(L(\frac12)\) \(\approx\) \(2.013824711\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-2.23 + 0.126i)T \)
11 \( 1 + T \)
good7 \( 1 - 3.92iT - 7T^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
17 \( 1 + 2.54iT - 17T^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 - 3.40T + 29T^{2} \)
31 \( 1 - 9.61T + 31T^{2} \)
37 \( 1 - 6.71iT - 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 + 2.14iT - 43T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 + 8.67iT - 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 7.78T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 1.38T + 71T^{2} \)
73 \( 1 - 5.75iT - 73T^{2} \)
79 \( 1 + 2.93T + 79T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 7.29iT - 97T^{2} \)
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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

−9.743045768422915568613318813672, −8.714278665513436324959575560299, −8.215058836178849497695674645175, −7.10379822866485175595653848203, −6.20804419935666646067876146811, −5.50899196929967628474438740208, −4.96616205915339478148728450095, −3.00941280973980306987487135668, −2.50848682678238450181739080292, −1.22403795735428968035805199733, 1.01464679219656282296837923192, 2.42216054001782342089826904279, 3.63945234530643998121188040464, 4.49402664709133296481132752198, 5.32351365341865917733483626986, 6.41719488195800304008597184975, 7.00945406811493526428105329665, 8.083151348584149043398392305116, 8.937055880380474458911304438130, 9.939144595814732887307770776854

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