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Properties

Label 2-2880-16.13-c1-0-2
Degree $2$
Conductor $2880$
Sign $-0.541 - 0.841i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
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  + (−0.707 + 0.707i)5-s − 4.35i·7-s + (−2.37 + 2.37i)11-s + (3.66 + 3.66i)13-s − 5.50·17-s + (−0.0623 − 0.0623i)19-s − 2.71i·23-s − 1.00i·25-s + (−2.48 − 2.48i)29-s + 6.29·31-s + (3.07 + 3.07i)35-s + (−6.02 + 6.02i)37-s + 1.43i·41-s + (−0.185 + 0.185i)43-s − 4.11·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s − 1.64i·7-s + (−0.715 + 0.715i)11-s + (1.01 + 1.01i)13-s − 1.33·17-s + (−0.0142 − 0.0142i)19-s − 0.565i·23-s − 0.200i·25-s + (−0.462 − 0.462i)29-s + 1.13·31-s + (0.520 + 0.520i)35-s + (−0.990 + 0.990i)37-s + 0.224i·41-s + (−0.0283 + 0.0283i)43-s − 0.599·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.541 - 0.841i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.541 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5900701263\)
\(L(\frac12)\) \(\approx\) \(0.5900701263\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 + (2.37 - 2.37i)T - 11iT^{2} \)
13 \( 1 + (-3.66 - 3.66i)T + 13iT^{2} \)
17 \( 1 + 5.50T + 17T^{2} \)
19 \( 1 + (0.0623 + 0.0623i)T + 19iT^{2} \)
23 \( 1 + 2.71iT - 23T^{2} \)
29 \( 1 + (2.48 + 2.48i)T + 29iT^{2} \)
31 \( 1 - 6.29T + 31T^{2} \)
37 \( 1 + (6.02 - 6.02i)T - 37iT^{2} \)
41 \( 1 - 1.43iT - 41T^{2} \)
43 \( 1 + (0.185 - 0.185i)T - 43iT^{2} \)
47 \( 1 + 4.11T + 47T^{2} \)
53 \( 1 + (9.16 - 9.16i)T - 53iT^{2} \)
59 \( 1 + (-5.17 + 5.17i)T - 59iT^{2} \)
61 \( 1 + (-7.00 - 7.00i)T + 61iT^{2} \)
67 \( 1 + (3.40 + 3.40i)T + 67iT^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + 6.32iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (-3.00 - 3.00i)T + 83iT^{2} \)
89 \( 1 - 4.12iT - 89T^{2} \)
97 \( 1 + 8.15T + 97T^{2} \)
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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

−8.949927328725632157312600312596, −8.189619647377552306485533389110, −7.45832610948514902311527280879, −6.73213279466706697248507377362, −6.36013207497723240435843144547, −4.83166932883856440618667637275, −4.32060785858711167687892297424, −3.63286514853493173283676010496, −2.43114448474268240714192324117, −1.26126068046531363066251731519, 0.19091902473264653294689789370, 1.79963556242019815614547829224, 2.82493630004815003197404046325, 3.54109164195731325318701257835, 4.78121190517589120262968170646, 5.55208567965059300954914782530, 5.97594111056115027905762998846, 6.96347015551883845051655448019, 8.146238632154316213804775041919, 8.461117680202485411736644235335

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