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Properties

Label 2-2880-48.11-c1-0-13
Degree $2$
Conductor $2880$
Sign $0.943 - 0.331i$
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 + 3.01·7-s + (−2.52 − 2.52i)11-s + (0.848 − 0.848i)13-s + 8.11i·17-s + (−1.76 − 1.76i)19-s + 8.68i·23-s − 1.00i·25-s + (5.71 + 5.71i)29-s + 1.57i·31-s + (2.13 − 2.13i)35-s + (4.56 + 4.56i)37-s + 11.3·41-s + (4.33 − 4.33i)43-s − 7.46·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + 1.13·7-s + (−0.759 − 0.759i)11-s + (0.235 − 0.235i)13-s + 1.96i·17-s + (−0.405 − 0.405i)19-s + 1.81i·23-s − 0.200i·25-s + (1.06 + 1.06i)29-s + 0.282i·31-s + (0.360 − 0.360i)35-s + (0.749 + 0.749i)37-s + 1.77·41-s + (0.661 − 0.661i)43-s − 1.08·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166251678\)
\(L(\frac12)\) \(\approx\) \(2.166251678\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.01T + 7T^{2} \)
11 \( 1 + (2.52 + 2.52i)T + 11iT^{2} \)
13 \( 1 + (-0.848 + 0.848i)T - 13iT^{2} \)
17 \( 1 - 8.11iT - 17T^{2} \)
19 \( 1 + (1.76 + 1.76i)T + 19iT^{2} \)
23 \( 1 - 8.68iT - 23T^{2} \)
29 \( 1 + (-5.71 - 5.71i)T + 29iT^{2} \)
31 \( 1 - 1.57iT - 31T^{2} \)
37 \( 1 + (-4.56 - 4.56i)T + 37iT^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + (-4.33 + 4.33i)T - 43iT^{2} \)
47 \( 1 + 7.46T + 47T^{2} \)
53 \( 1 + (-4.02 + 4.02i)T - 53iT^{2} \)
59 \( 1 + (2.67 + 2.67i)T + 59iT^{2} \)
61 \( 1 + (-5.84 + 5.84i)T - 61iT^{2} \)
67 \( 1 + (-8.64 - 8.64i)T + 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 - 2.42iT - 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 + (4.26 - 4.26i)T - 83iT^{2} \)
89 \( 1 + 3.00T + 89T^{2} \)
97 \( 1 + 7.39T + 97T^{2} \)
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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

−8.460571263586802934280988763613, −8.313244744011025502515867349665, −7.50708788809943524982979649447, −6.38442403999512207313330077037, −5.65480949435790611102674998034, −5.05672031836190086580206978148, −4.12927169542815515415475980679, −3.17375314093907293014139450959, −1.97439582417648264570871346931, −1.11532880254053141543609741006, 0.805890219985728519233506921877, 2.30690849215592462227040956197, 2.64263277730567290039517558631, 4.35045756179251485123803025711, 4.66087766986501955863010754309, 5.62228692035132415992726945224, 6.48447591501455099989487271130, 7.34008423285244988762538725395, 7.914603337067154373495338951552, 8.644212100404363541954104855677

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