Properties

Label 2-5040-21.20-c1-0-51
Degree $2$
Conductor $5040$
Sign $-0.836 + 0.547i$
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  − 5-s + (−0.0951 + 2.64i)7-s − 5.28i·11-s + 2.19i·13-s + 1.04·17-s − 6.43i·19-s − 7.47i·23-s + 25-s + 7.47i·29-s + 9.09i·31-s + (0.0951 − 2.64i)35-s − 0.855·37-s − 2.19·41-s + 0.954·43-s − 11.0·47-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.0359 + 0.999i)7-s − 1.59i·11-s + 0.607i·13-s + 0.253·17-s − 1.47i·19-s − 1.55i·23-s + 0.200·25-s + 1.38i·29-s + 1.63i·31-s + (0.0160 − 0.446i)35-s − 0.140·37-s − 0.342·41-s + 0.145·43-s − 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.836 + 0.547i$
Analytic conductor: \(40.2446\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5040,\ (\ :1/2),\ -0.836 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4413896154\)
\(L(\frac12)\) \(\approx\) \(0.4413896154\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (0.0951 - 2.64i)T \)
good11 \( 1 + 5.28iT - 11T^{2} \)
13 \( 1 - 2.19iT - 13T^{2} \)
17 \( 1 - 1.04T + 17T^{2} \)
19 \( 1 + 6.43iT - 19T^{2} \)
23 \( 1 + 7.47iT - 23T^{2} \)
29 \( 1 - 7.47iT - 29T^{2} \)
31 \( 1 - 9.09iT - 31T^{2} \)
37 \( 1 + 0.855T + 37T^{2} \)
41 \( 1 + 2.19T + 41T^{2} \)
43 \( 1 - 0.954T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 3.09iT - 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 5.33T + 67T^{2} \)
71 \( 1 - 6.43iT - 71T^{2} \)
73 \( 1 - 4.57iT - 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + 4.38T + 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−8.321488228754995380761214920814, −6.91296433261602343972345568310, −6.73587007114032601103835795234, −5.67010615748649237727142520222, −5.10080844863234680826946994031, −4.25685383281580170905843231439, −3.11524479789980278047727264052, −2.78419743312902339519311792915, −1.40504303783983409760650919522, −0.12319444385561396371680421981, 1.26720492413828452035832569673, 2.19266821540687892750539315141, 3.49324481666117236229477623632, 3.98370186592904489465328207127, 4.74018045702458173166755507005, 5.61928165390656916164256966551, 6.43480578340247466277690654954, 7.35803816104168685381480749773, 7.71996349946965925555746278875, 8.182217301881173607146699625663

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