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Properties

Label 2-2880-16.13-c1-0-13
Degree $2$
Conductor $2880$
Sign $0.910 - 0.412i$
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 − 1.73i·7-s + (0.505 − 0.505i)11-s + (−1.88 − 1.88i)13-s − 4.53·17-s + (3.22 + 3.22i)19-s + 8.85i·23-s − 1.00i·25-s + (2.44 + 2.44i)29-s + 5.70·31-s + (1.22 + 1.22i)35-s + (−5.35 + 5.35i)37-s − 10.0i·41-s + (2.10 − 2.10i)43-s + 4.32·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s − 0.656i·7-s + (0.152 − 0.152i)11-s + (−0.523 − 0.523i)13-s − 1.09·17-s + (0.738 + 0.738i)19-s + 1.84i·23-s − 0.200i·25-s + (0.453 + 0.453i)29-s + 1.02·31-s + (0.207 + 0.207i)35-s + (−0.880 + 0.880i)37-s − 1.56i·41-s + (0.321 − 0.321i)43-s + 0.630·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.910 - 0.412i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.514716596\)
\(L(\frac12)\) \(\approx\) \(1.514716596\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + (-0.505 + 0.505i)T - 11iT^{2} \)
13 \( 1 + (1.88 + 1.88i)T + 13iT^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + (-3.22 - 3.22i)T + 19iT^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 + (-2.44 - 2.44i)T + 29iT^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + (5.35 - 5.35i)T - 37iT^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (-2.10 + 2.10i)T - 43iT^{2} \)
47 \( 1 - 4.32T + 47T^{2} \)
53 \( 1 + (-1.37 + 1.37i)T - 53iT^{2} \)
59 \( 1 + (-6.64 + 6.64i)T - 59iT^{2} \)
61 \( 1 + (-5.26 - 5.26i)T + 61iT^{2} \)
67 \( 1 + (-10.5 - 10.5i)T + 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + 6.63iT - 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 + (-9.15 - 9.15i)T + 83iT^{2} \)
89 \( 1 + 3.23iT - 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−8.770859334900572856678402049482, −7.980935102345099222621907061640, −7.26764851558440355198492269479, −6.77169730309990484903209178584, −5.69541584328655985025156466888, −4.97906023408319339544955797226, −3.91789091229275338888275798879, −3.34390822318326289725520575670, −2.18191738594498720835375407124, −0.872608034208744197971800362498, 0.64946134826685848737088672857, 2.18851946516140603382024544798, 2.83294359107631061236270559824, 4.24774181202237598059680256246, 4.66126576094150137691779392609, 5.60412031288861845154145224977, 6.60667447214444559136946598717, 7.04803450579820980283693072513, 8.176589174385742989000653576411, 8.682848627236624632502758475345

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