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Properties

Label 2-2880-120.77-c1-0-10
Degree $2$
Conductor $2880$
Sign $-0.279 - 0.960i$
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  + (−2.22 − 0.224i)5-s + (1.41 + 1.41i)7-s − 3.46·11-s + (0.317 + 0.317i)13-s + (2.04 − 2.04i)17-s + 2.89·19-s + (0.449 + 0.449i)23-s + (4.89 + i)25-s − 3.55i·29-s + 4.09·31-s + (−2.82 − 3.46i)35-s + (−5.97 + 5.97i)37-s − 1.41i·41-s + (−2.89 − 2.89i)43-s + (−6.44 + 6.44i)47-s + ⋯
L(s)  = 1  + (−0.994 − 0.100i)5-s + (0.534 + 0.534i)7-s − 1.04·11-s + (0.0881 + 0.0881i)13-s + (0.497 − 0.497i)17-s + 0.665·19-s + (0.0937 + 0.0937i)23-s + (0.979 + 0.200i)25-s − 0.659i·29-s + 0.736·31-s + (−0.478 − 0.585i)35-s + (−0.982 + 0.982i)37-s − 0.220i·41-s + (−0.442 − 0.442i)43-s + (−0.940 + 0.940i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.279 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9166658459\)
\(L(\frac12)\) \(\approx\) \(0.9166658459\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.224i)T \)
good7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + (-0.317 - 0.317i)T + 13iT^{2} \)
17 \( 1 + (-2.04 + 2.04i)T - 17iT^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 + (-0.449 - 0.449i)T + 23iT^{2} \)
29 \( 1 + 3.55iT - 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + (5.97 - 5.97i)T - 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + (2.89 + 2.89i)T + 43iT^{2} \)
47 \( 1 + (6.44 - 6.44i)T - 47iT^{2} \)
53 \( 1 + (2.89 - 2.89i)T - 53iT^{2} \)
59 \( 1 - 6.29iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + (7.89 - 7.89i)T - 73iT^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + (-9.12 + 9.12i)T - 83iT^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 + (7.89 + 7.89i)T + 97iT^{2} \)
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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

−8.710094689571685579985122884071, −8.269753434478260205880081437590, −7.57271407023922114036090277725, −6.94946214922241898742999449526, −5.75417479668757105777358195601, −5.09195011655027384952611921297, −4.39958985888161451309470682805, −3.29700434018604448015085969143, −2.56508992228305282936266940677, −1.14681968529004822680441250303, 0.32859602538284922613803682345, 1.65930967806847802974409632254, 3.05480325554509102891535347118, 3.65972620689776834457503534092, 4.74520616116004175552560972090, 5.20701505239600087119402720307, 6.38456597596219306353916177759, 7.22952541892673043049360764202, 7.924864764075778521492292019444, 8.207022894435963225085208084043

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