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Properties

Label 2-2880-48.35-c1-0-29
Degree $2$
Conductor $2880$
Sign $-0.430 + 0.902i$
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 − 0.841·7-s + (3.43 − 3.43i)11-s + (−0.690 − 0.690i)13-s − 0.821i·17-s + (3.35 − 3.35i)19-s − 0.473i·23-s + 1.00i·25-s + (−0.820 + 0.820i)29-s + 2.43i·31-s + (0.595 + 0.595i)35-s + (−6.22 + 6.22i)37-s − 3.45·41-s + (−3.26 − 3.26i)43-s + 0.936·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 0.318·7-s + (1.03 − 1.03i)11-s + (−0.191 − 0.191i)13-s − 0.199i·17-s + (0.770 − 0.770i)19-s − 0.0987i·23-s + 0.200i·25-s + (−0.152 + 0.152i)29-s + 0.436i·31-s + (0.100 + 0.100i)35-s + (−1.02 + 1.02i)37-s − 0.539·41-s + (−0.497 − 0.497i)43-s + 0.136·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.430 + 0.902i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.236693163\)
\(L(\frac12)\) \(\approx\) \(1.236693163\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.841T + 7T^{2} \)
11 \( 1 + (-3.43 + 3.43i)T - 11iT^{2} \)
13 \( 1 + (0.690 + 0.690i)T + 13iT^{2} \)
17 \( 1 + 0.821iT - 17T^{2} \)
19 \( 1 + (-3.35 + 3.35i)T - 19iT^{2} \)
23 \( 1 + 0.473iT - 23T^{2} \)
29 \( 1 + (0.820 - 0.820i)T - 29iT^{2} \)
31 \( 1 - 2.43iT - 31T^{2} \)
37 \( 1 + (6.22 - 6.22i)T - 37iT^{2} \)
41 \( 1 + 3.45T + 41T^{2} \)
43 \( 1 + (3.26 + 3.26i)T + 43iT^{2} \)
47 \( 1 - 0.936T + 47T^{2} \)
53 \( 1 + (-6.73 - 6.73i)T + 53iT^{2} \)
59 \( 1 + (-9.53 + 9.53i)T - 59iT^{2} \)
61 \( 1 + (6.32 + 6.32i)T + 61iT^{2} \)
67 \( 1 + (2.77 - 2.77i)T - 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + 6.65iT - 73T^{2} \)
79 \( 1 + 15.9iT - 79T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−8.689259557872758620728483723764, −7.82087358025655631549076400220, −6.94110613902056248101283881099, −6.35344956408021907024554344421, −5.38204344899901397984435527920, −4.67809977512540098667794021317, −3.56325345616354414397137908749, −3.06510317282263197627587292258, −1.56504556167846386952036081347, −0.41565794877265864524768233136, 1.34680147640336479775223306608, 2.39520344589267727699148915127, 3.60401809981549229273301382985, 4.10190497719032769643804264179, 5.15928026507990239495760851544, 6.02980638762576956959654180587, 6.94380175213606839152513292032, 7.29051756808923210943334816478, 8.265515597664111169334279009106, 9.051704493784961521595372511295

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