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Properties

Label 2-5040-5.4-c1-0-36
Degree $2$
Conductor $5040$
Sign $0.241 - 0.970i$
Analytic cond. $40.2446$
Root an. cond. $6.34386$
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  + (2.17 + 0.539i)5-s + i·7-s + 5.41·11-s + 4.34i·13-s + 1.07i·17-s + 4.34·19-s + 6.34i·23-s + (4.41 + 2.34i)25-s − 8.83·29-s + 4.34·31-s + (−0.539 + 2.17i)35-s + 8.68i·37-s − 8.34·41-s − 6.15i·43-s + 6.83i·47-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)5-s + 0.377i·7-s + 1.63·11-s + 1.20i·13-s + 0.261i·17-s + 0.995·19-s + 1.32i·23-s + (0.883 + 0.468i)25-s − 1.64·29-s + 0.779·31-s + (−0.0911 + 0.366i)35-s + 1.42i·37-s − 1.30·41-s − 0.938i·43-s + 0.997i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.638906129\)
\(L(\frac12)\) \(\approx\) \(2.638906129\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.539i)T \)
7 \( 1 - iT \)
good11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 - 4.34iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 - 4.34T + 19T^{2} \)
23 \( 1 - 6.34iT - 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 - 4.34T + 31T^{2} \)
37 \( 1 - 8.68iT - 37T^{2} \)
41 \( 1 + 8.34T + 41T^{2} \)
43 \( 1 + 6.15iT - 43T^{2} \)
47 \( 1 - 6.83iT - 47T^{2} \)
53 \( 1 + 6.18iT - 53T^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 0.680T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 - 6.49T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−8.637389190950124826468984724803, −7.50074496944604088833127390291, −6.82758198713030061528495421114, −6.27119531021125502339769991617, −5.59805946261379186369097042657, −4.78278414054773851838106802743, −3.80530723647412184223168735575, −3.11292514693152223794055164383, −1.80008523886097250863541947771, −1.45849441114014214021733887501, 0.72174761720976282092262123330, 1.55799116004474242025304180930, 2.64387412583295144009033824384, 3.55639171219658295957004474112, 4.37360887843136529811928138406, 5.29937619009144064015935405938, 5.86720930232803946798648471332, 6.64360979679106640009537990562, 7.24507833932134258284056302769, 8.148599214790657913645503567989

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