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Properties

Label 2-5040-28.27-c1-0-16
Degree $2$
Conductor $5040$
Sign $-0.679 - 0.733i$
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  + i·5-s + (0.585 + 2.58i)7-s − 3.02i·11-s + 0.829i·13-s + 0.756i·17-s − 2.22·19-s + 0.0727i·23-s − 25-s + 4.71·29-s + 0.264·31-s + (−2.58 + 0.585i)35-s − 4.13·37-s + 7.42i·41-s + 9.01i·43-s − 3.88·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + (0.221 + 0.975i)7-s − 0.910i·11-s + 0.229i·13-s + 0.183i·17-s − 0.509·19-s + 0.0151i·23-s − 0.200·25-s + 0.876·29-s + 0.0475·31-s + (−0.436 + 0.0989i)35-s − 0.680·37-s + 1.15i·41-s + 1.37i·43-s − 0.567·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (4591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ -0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237125370\)
\(L(\frac12)\) \(\approx\) \(1.237125370\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-0.585 - 2.58i)T \)
good11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 - 0.829iT - 13T^{2} \)
17 \( 1 - 0.756iT - 17T^{2} \)
19 \( 1 + 2.22T + 19T^{2} \)
23 \( 1 - 0.0727iT - 23T^{2} \)
29 \( 1 - 4.71T + 29T^{2} \)
31 \( 1 - 0.264T + 31T^{2} \)
37 \( 1 + 4.13T + 37T^{2} \)
41 \( 1 - 7.42iT - 41T^{2} \)
43 \( 1 - 9.01iT - 43T^{2} \)
47 \( 1 + 3.88T + 47T^{2} \)
53 \( 1 - 4.40T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 0.745iT - 61T^{2} \)
67 \( 1 - 4.48iT - 67T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 - 7.24iT - 73T^{2} \)
79 \( 1 - 2.57iT - 79T^{2} \)
83 \( 1 + 4.52T + 83T^{2} \)
89 \( 1 - 9.82iT - 89T^{2} \)
97 \( 1 - 17.5iT - 97T^{2} \)
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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

−8.278230723148997285330700415092, −8.128200456921143688339593955968, −6.87828008633229752048782426737, −6.36136127643861203157150673949, −5.66708874676000687698537765808, −4.92314831205583935648554338942, −3.98854124159595961067988483840, −3.04131883823730423343034075572, −2.41614629838143942806403119829, −1.27943825696674243719660130528, 0.33718222555137846325959125048, 1.47166609913581779876132949070, 2.39763140997362316452896124935, 3.60822886757755361217756294332, 4.27421186994400670638578691187, 4.95126332862157028518943498248, 5.67704347834159787716317743708, 6.84083783106593473073711987885, 7.09855574305040221889946161059, 8.013562413344282832535035077648

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