Properties

Label 2-2888-152.139-c0-0-3
Degree $2$
Conductor $2888$
Sign $0.499 + 0.866i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (−0.499 − 0.866i)27-s + (−0.939 + 0.342i)32-s + (0.173 + 0.984i)33-s + (0.347 − 1.96i)34-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)24-s + (−0.939 + 0.342i)25-s + (−0.499 − 0.866i)27-s + (−0.939 + 0.342i)32-s + (0.173 + 0.984i)33-s + (0.347 − 1.96i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.556180093\)
\(L(\frac12)\) \(\approx\) \(2.556180093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.314214085365305114729469506272, −8.033632113141193869270454077821, −7.40135665204493632844512928932, −6.40558725268815130584000551280, −5.57277645709845277665282392038, −4.71052538859762984216868679588, −3.93525383357316514256716777302, −3.16788614294526589651645975774, −2.49061118893622838472850498626, −1.31871239397045894815729063763, 1.69878754120843026987059563009, 2.80295473625207870785760040799, 3.56087509032473436438334635633, 4.14482564202360871784907291852, 5.54610683070887019395640178291, 5.85170755722400125605240723778, 6.83432412733017173713425009291, 7.74027043131350287857904363801, 8.109179140670212054212320198698, 8.749090972671818670668008080098

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