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Properties

Label 2-2880-120.59-c1-0-40
Degree $2$
Conductor $2880$
Sign $-0.492 + 0.870i$
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  + (1.41 − 1.73i)5-s − 2.44·7-s + 1.41i·11-s + 2.44·13-s + 3.46·17-s − 6·19-s − 6.92i·23-s + (−0.999 − 4.89i)25-s − 5.65·29-s − 2i·31-s + (−3.46 + 4.24i)35-s + 7.34·37-s + 4.24i·41-s − 4.89i·43-s + 3.46i·47-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)5-s − 0.925·7-s + 0.426i·11-s + 0.679·13-s + 0.840·17-s − 1.37·19-s − 1.44i·23-s + (−0.199 − 0.979i)25-s − 1.05·29-s − 0.359i·31-s + (−0.585 + 0.717i)35-s + 1.20·37-s + 0.662i·41-s − 0.747i·43-s + 0.505i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.251887115\)
\(L(\frac12)\) \(\approx\) \(1.251887115\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.73i)T \)
good7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 - 1.41iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 + 14.6iT - 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−8.490968671999708611765681326108, −8.004816041957478665985520166754, −6.75652985831651551362811363146, −6.26403208700761464210796069283, −5.54764788401589656486505218744, −4.56598561405165121959681376722, −3.83456734596519563256609939459, −2.68855097424618587932169025292, −1.72121674956024784836165101108, −0.39298305501320008442952426042, 1.37371967150443383049790152902, 2.56385910816753463458749609008, 3.37605512447062917723233081208, 4.05593911444364017343426640665, 5.55296957884204587785159977306, 5.92817678000346821411711224260, 6.67623851828959645658933119166, 7.39952575028413481733722786295, 8.275286523466054910422363560083, 9.217750571604894156703453432770

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