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Properties

Label 2-2880-48.35-c1-0-10
Degree $2$
Conductor $2880$
Sign $-0.220 - 0.975i$
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 + 2.87·7-s + (−2.03 + 2.03i)11-s + (3.87 + 3.87i)13-s + 1.41i·17-s + (−5.87 + 5.87i)19-s − 5.47i·23-s + 1.00i·25-s + (−2.82 + 2.82i)29-s + 4i·31-s + (2.03 + 2.03i)35-s + (−1 + i)37-s − 11.1·41-s + (6 + 6i)43-s − 1.41·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.08·7-s + (−0.612 + 0.612i)11-s + (1.07 + 1.07i)13-s + 0.342i·17-s + (−1.34 + 1.34i)19-s − 1.14i·23-s + 0.200i·25-s + (−0.525 + 0.525i)29-s + 0.718i·31-s + (0.343 + 0.343i)35-s + (−0.164 + 0.164i)37-s − 1.73·41-s + (0.914 + 0.914i)43-s − 0.206·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771290351\)
\(L(\frac12)\) \(\approx\) \(1.771290351\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.87T + 7T^{2} \)
11 \( 1 + (2.03 - 2.03i)T - 11iT^{2} \)
13 \( 1 + (-3.87 - 3.87i)T + 13iT^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 + (5.87 - 5.87i)T - 19iT^{2} \)
23 \( 1 + 5.47iT - 23T^{2} \)
29 \( 1 + (2.82 - 2.82i)T - 29iT^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + 53iT^{2} \)
59 \( 1 + (-9.10 + 9.10i)T - 59iT^{2} \)
61 \( 1 + (10.7 + 10.7i)T + 61iT^{2} \)
67 \( 1 + (3.12 - 3.12i)T - 67iT^{2} \)
71 \( 1 - 9.71iT - 71T^{2} \)
73 \( 1 + 5.74iT - 73T^{2} \)
79 \( 1 + 3.74iT - 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 + 5.83T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.673713824523213924503226287624, −8.422785893987203987866259532599, −7.54038229767094062063688032426, −6.59594841292395955564383269877, −6.08531321991016783315599786119, −5.00380475629689170944529921602, −4.36932171344996419199912157326, −3.45890569073583892046864070935, −2.05838559730773318205032418357, −1.61189084826770798332564992977, 0.54034849509808935714037829351, 1.74157712956026215346546660869, 2.74944655075850867585038899453, 3.81521270567618420811176800521, 4.76771677642062997928706064232, 5.50145919672086356942812317452, 6.04020850455960833388060102559, 7.18447633989395312599110064236, 7.931954629073460745265986627570, 8.577040165777209160767830324705

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