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Properties

Label 2-2880-120.53-c1-0-8
Degree $2$
Conductor $2880$
Sign $-0.680 - 0.732i$
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  + (1.27 + 1.83i)5-s + (0.254 − 0.254i)7-s − 1.83·11-s + (0.689 − 0.689i)13-s + (2.00 + 2.00i)17-s − 4.78·19-s + (−4.64 + 4.64i)23-s + (−1.73 + 4.68i)25-s + 1.83i·29-s + 3.61·31-s + (0.791 + 0.141i)35-s + (−5.09 − 5.09i)37-s − 0.516i·41-s + (−4.20 + 4.20i)43-s + (−1.22 − 1.22i)47-s + ⋯
L(s)  = 1  + (0.571 + 0.820i)5-s + (0.0960 − 0.0960i)7-s − 0.552·11-s + (0.191 − 0.191i)13-s + (0.486 + 0.486i)17-s − 1.09·19-s + (−0.968 + 0.968i)23-s + (−0.346 + 0.937i)25-s + 0.340i·29-s + 0.649·31-s + (0.133 + 0.0239i)35-s + (−0.837 − 0.837i)37-s − 0.0806i·41-s + (−0.640 + 0.640i)43-s + (−0.178 − 0.178i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.680 - 0.732i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188303800\)
\(L(\frac12)\) \(\approx\) \(1.188303800\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.83i)T \)
good7 \( 1 + (-0.254 + 0.254i)T - 7iT^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 + (-0.689 + 0.689i)T - 13iT^{2} \)
17 \( 1 + (-2.00 - 2.00i)T + 17iT^{2} \)
19 \( 1 + 4.78T + 19T^{2} \)
23 \( 1 + (4.64 - 4.64i)T - 23iT^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 - 3.61T + 31T^{2} \)
37 \( 1 + (5.09 + 5.09i)T + 37iT^{2} \)
41 \( 1 + 0.516iT - 41T^{2} \)
43 \( 1 + (4.20 - 4.20i)T - 43iT^{2} \)
47 \( 1 + (1.22 + 1.22i)T + 47iT^{2} \)
53 \( 1 + (-3.36 - 3.36i)T + 53iT^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 - 2.95iT - 61T^{2} \)
67 \( 1 + (6.25 + 6.25i)T + 67iT^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 + (-7.15 - 7.15i)T + 73iT^{2} \)
79 \( 1 + 6.76iT - 79T^{2} \)
83 \( 1 + (-0.933 - 0.933i)T + 83iT^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + (7.21 - 7.21i)T - 97iT^{2} \)
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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

−9.102347804658794631702326176635, −8.182040195034574296140586853941, −7.58286571346108147764971639698, −6.72904855379910057258578661600, −5.97863433222269592767487086559, −5.41397328879682535829950074299, −4.25699281028844893532865280026, −3.38632046519945868712514211431, −2.46927401131639789756038253116, −1.52640398432318929842044261840, 0.35288524853957406582606391864, 1.73868026941301973250272084112, 2.54605564684779196200691761661, 3.80569121591346417099944791081, 4.72566002587688949890822132090, 5.30506094827673557852108802046, 6.21483058766651183946660332569, 6.81717897900396496342613168536, 8.141835668663526662413541051190, 8.309624512647044115646291525999

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