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Properties

Label 2-2880-48.35-c1-0-7
Degree $2$
Conductor $2880$
Sign $-0.459 - 0.887i$
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 − 1.61·7-s + (−0.222 + 0.222i)11-s + (2.23 + 2.23i)13-s + 2.72i·17-s + (−1.00 + 1.00i)19-s − 1.97i·23-s + 1.00i·25-s + (1.27 − 1.27i)29-s + 2.63i·31-s + (−1.14 − 1.14i)35-s + (−1.18 + 1.18i)37-s − 0.870·41-s + (−3.10 − 3.10i)43-s − 6.36·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s − 0.610·7-s + (−0.0669 + 0.0669i)11-s + (0.620 + 0.620i)13-s + 0.659i·17-s + (−0.231 + 0.231i)19-s − 0.412i·23-s + 0.200i·25-s + (0.237 − 0.237i)29-s + 0.473i·31-s + (−0.193 − 0.193i)35-s + (−0.194 + 0.194i)37-s − 0.135·41-s + (−0.473 − 0.473i)43-s − 0.928·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.459 - 0.887i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.459 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.177108894\)
\(L(\frac12)\) \(\approx\) \(1.177108894\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + (0.222 - 0.222i)T - 11iT^{2} \)
13 \( 1 + (-2.23 - 2.23i)T + 13iT^{2} \)
17 \( 1 - 2.72iT - 17T^{2} \)
19 \( 1 + (1.00 - 1.00i)T - 19iT^{2} \)
23 \( 1 + 1.97iT - 23T^{2} \)
29 \( 1 + (-1.27 + 1.27i)T - 29iT^{2} \)
31 \( 1 - 2.63iT - 31T^{2} \)
37 \( 1 + (1.18 - 1.18i)T - 37iT^{2} \)
41 \( 1 + 0.870T + 41T^{2} \)
43 \( 1 + (3.10 + 3.10i)T + 43iT^{2} \)
47 \( 1 + 6.36T + 47T^{2} \)
53 \( 1 + (-0.945 - 0.945i)T + 53iT^{2} \)
59 \( 1 + (8.08 - 8.08i)T - 59iT^{2} \)
61 \( 1 + (-10.5 - 10.5i)T + 61iT^{2} \)
67 \( 1 + (-2.50 + 2.50i)T - 67iT^{2} \)
71 \( 1 + 3.39iT - 71T^{2} \)
73 \( 1 - 15.2iT - 73T^{2} \)
79 \( 1 - 8.65iT - 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
97 \( 1 + 6.56T + 97T^{2} \)
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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

−8.896904026046864663403983590615, −8.444315144812972177138692504066, −7.42134390888142649695983988142, −6.57196388916279291006661979609, −6.19051274242430392370227039121, −5.22779235411560470857836178646, −4.19323031827640864271561382731, −3.43604147318495900488899944528, −2.44602337406663407735900934844, −1.37328624635063484168714031141, 0.37409692400820137494463859429, 1.68052376343573342930666094912, 2.89303603151714779716607520902, 3.60049621542029542116540913111, 4.72623307157023589465461434913, 5.44271763639142235709383224816, 6.27482892751837712083192064023, 6.88654991604453159619784857131, 7.913342692755049525418771074242, 8.477405379124275975793484500634

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