Properties

Label 2-5040-5.4-c1-0-47
Degree $2$
Conductor $5040$
Sign $0.749 + 0.662i$
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  + (−1.48 + 1.67i)5-s + i·7-s + 0.387·11-s − 2.96i·13-s + 3.35i·17-s − 2.96·19-s − 0.962i·23-s + (−0.612 − 4.96i)25-s + 1.22·29-s − 2.96·31-s + (−1.67 − 1.48i)35-s − 5.92i·37-s − 1.03·41-s − 10.7i·43-s − 3.22i·47-s + ⋯
L(s)  = 1  + (−0.662 + 0.749i)5-s + 0.377i·7-s + 0.116·11-s − 0.821i·13-s + 0.812i·17-s − 0.679·19-s − 0.200i·23-s + (−0.122 − 0.992i)25-s + 0.227·29-s − 0.532·31-s + (−0.283 − 0.250i)35-s − 0.974i·37-s − 0.162·41-s − 1.63i·43-s − 0.470i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.749 + 0.662i$
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: \((2,\ 5040,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.153934384\)
\(L(\frac12)\) \(\approx\) \(1.153934384\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 1.67i)T \)
7 \( 1 - iT \)
good11 \( 1 - 0.387T + 11T^{2} \)
13 \( 1 + 2.96iT - 13T^{2} \)
17 \( 1 - 3.35iT - 17T^{2} \)
19 \( 1 + 2.96T + 19T^{2} \)
23 \( 1 + 0.962iT - 23T^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 + 5.92iT - 37T^{2} \)
41 \( 1 + 1.03T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + 3.22iT - 47T^{2} \)
53 \( 1 - 5.66iT - 53T^{2} \)
59 \( 1 + 3.22T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 - 6.18iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 3.22iT - 83T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 - 7.73iT - 97T^{2} \)
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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

−8.304676797905347284150352224852, −7.35683412225918839775100142685, −6.83442638872407963500223672436, −5.96871289247508842849324494015, −5.37774239035769092835894765483, −4.23745412274369756692447816449, −3.67286586839929888059526103492, −2.78598502913500036042658295919, −1.94245463844161543277341908029, −0.39496115836632745959942262345, 0.859945658357297931804624746460, 1.88071727369130462449103292935, 3.08336073777974469444590746386, 3.96257363586955074180395428124, 4.60324614554178138910536370500, 5.19283716061223600030351701222, 6.27879721488225893325101772753, 6.92269990339413392860888827580, 7.66701419936897244927441736016, 8.289379093269366632569198674236

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