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Properties

Label 2-4080-5.4-c1-0-17
Degree $2$
Conductor $4080$
Sign $-0.120 - 0.992i$
Analytic cond. $32.5789$
Root an. cond. $5.70779$
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 + (−0.269 − 2.21i)5-s + 1.32i·7-s − 9-s − 1.57·11-s − 4.68i·13-s + (2.21 − 0.269i)15-s + i·17-s + 0.424·19-s − 1.32·21-s + 5.71i·23-s + (−4.85 + 1.19i)25-s − i·27-s − 6.01·29-s − 3.05·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.120 − 0.992i)5-s + 0.502i·7-s − 0.333·9-s − 0.474·11-s − 1.29i·13-s + (0.573 − 0.0695i)15-s + 0.242i·17-s + 0.0974·19-s − 0.290·21-s + 1.19i·23-s + (−0.970 + 0.239i)25-s − 0.192i·27-s − 1.11·29-s − 0.549·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4080\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.120 - 0.992i$
Analytic conductor: \(32.5789\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4080} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4080,\ (\ :1/2),\ -0.120 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.052620360\)
\(L(\frac12)\) \(\approx\) \(1.052620360\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.269 + 2.21i)T \)
17 \( 1 - iT \)
good7 \( 1 - 1.32iT - 7T^{2} \)
11 \( 1 + 1.57T + 11T^{2} \)
13 \( 1 + 4.68iT - 13T^{2} \)
19 \( 1 - 0.424T + 19T^{2} \)
23 \( 1 - 5.71iT - 23T^{2} \)
29 \( 1 + 6.01T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 - 1.16iT - 37T^{2} \)
41 \( 1 - 9.43T + 41T^{2} \)
43 \( 1 - 4.16iT - 43T^{2} \)
47 \( 1 - 0.0211iT - 47T^{2} \)
53 \( 1 - 8.97iT - 53T^{2} \)
59 \( 1 - 3.92T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 - 7.27iT - 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 3.84iT - 73T^{2} \)
79 \( 1 - 5.36T + 79T^{2} \)
83 \( 1 + 1.31iT - 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 - 3.08iT - 97T^{2} \)
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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

−8.764168324287193209440098836183, −7.78939816656853882736543474817, −7.60780563842858802934532849550, −6.05054569163223254816236584093, −5.54591366262848343371200994723, −5.03726911858401859411577290655, −4.07610472526533506570087944049, −3.31264221646251593316358030224, −2.30221484319210245366523201323, −1.03158450326910915592044430696, 0.33254906311288942855496733400, 1.86731310317993285052102498554, 2.54223744854755787544448537465, 3.59335577004731217604821135281, 4.30261666501395466336074294730, 5.35303767315352757433012990608, 6.29901005985639456344300428214, 6.79653646971864406339539421254, 7.45936672121267622131765571321, 7.954176917233122014562558028850

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