Properties

Label 2-2888-152.131-c0-0-4
Degree $2$
Conductor $2888$
Sign $0.813 + 0.581i$
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  + (0.173 − 0.984i)2-s + (1.17 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (0.233 + 1.32i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + 1.34·18-s + (−1.43 − 1.20i)22-s + (−1.43 + 0.524i)24-s + (0.766 − 0.642i)25-s + (−0.266 + 0.460i)27-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (1.17 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (0.233 + 1.32i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + 1.34·18-s + (−1.43 − 1.20i)22-s + (−1.43 + 0.524i)24-s + (0.766 − 0.642i)25-s + (−0.266 + 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.898368884\)
\(L(\frac12)\) \(\approx\) \(1.898368884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
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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

−8.899989715096337851940710935306, −8.609609104569426836445396083750, −7.84524197949401456520881581676, −6.36053211547273937746630940375, −5.65421471222022446086069859236, −4.45924373153766730608235898900, −3.98585870723683865255302232045, −3.22589506891299867878157489033, −2.60778618228833295019566931744, −1.27586851317820056133708197222, 1.35267881924331781436672777891, 2.46222792121493171702326681119, 3.45495230969651361557952023987, 4.38574192095224667266831788334, 5.15597203837535785034156849401, 6.37679197483382470330448627073, 7.05641943212562433989150434623, 7.31137452733484196859698453034, 8.104367119965576496477951046497, 8.911042730049055672830226354554

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