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Properties

Label 2-2888-2888.2779-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.992 - 0.124i$
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  + (0.690 + 0.723i)2-s + (−0.788 + 0.712i)3-s + (−0.0459 + 0.998i)4-s + (−1.06 − 0.0780i)6-s + (−0.754 + 0.656i)8-s + (0.0131 − 0.129i)9-s + (0.870 + 0.846i)11-s + (−0.675 − 0.820i)12-s + (−0.995 − 0.0917i)16-s + (1.60 + 0.831i)17-s + (0.102 − 0.0797i)18-s + (0.280 + 0.959i)19-s + (−0.0111 + 1.21i)22-s + (0.126 − 1.05i)24-s + (−0.333 − 0.942i)25-s + ⋯
L(s)  = 1  + (0.690 + 0.723i)2-s + (−0.788 + 0.712i)3-s + (−0.0459 + 0.998i)4-s + (−1.06 − 0.0780i)6-s + (−0.754 + 0.656i)8-s + (0.0131 − 0.129i)9-s + (0.870 + 0.846i)11-s + (−0.675 − 0.820i)12-s + (−0.995 − 0.0917i)16-s + (1.60 + 0.831i)17-s + (0.102 − 0.0797i)18-s + (0.280 + 0.959i)19-s + (−0.0111 + 1.21i)22-s + (0.126 − 1.05i)24-s + (−0.333 − 0.942i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.992 - 0.124i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.992 - 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324595999\)
\(L(\frac12)\) \(\approx\) \(1.324595999\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.690 - 0.723i)T \)
19 \( 1 + (-0.280 - 0.959i)T \)
good3 \( 1 + (0.788 - 0.712i)T + (0.100 - 0.994i)T^{2} \)
5 \( 1 + (0.333 + 0.942i)T^{2} \)
7 \( 1 + (0.821 + 0.569i)T^{2} \)
11 \( 1 + (-0.870 - 0.846i)T + (0.0275 + 0.999i)T^{2} \)
13 \( 1 + (0.562 + 0.826i)T^{2} \)
17 \( 1 + (-1.60 - 0.831i)T + (0.577 + 0.816i)T^{2} \)
23 \( 1 + (0.912 + 0.410i)T^{2} \)
29 \( 1 + (0.703 - 0.710i)T^{2} \)
31 \( 1 + (-0.975 + 0.218i)T^{2} \)
37 \( 1 + (0.879 + 0.475i)T^{2} \)
41 \( 1 + (1.45 + 1.36i)T + (0.0642 + 0.997i)T^{2} \)
43 \( 1 + (-0.435 - 1.10i)T + (-0.729 + 0.684i)T^{2} \)
47 \( 1 + (0.367 + 0.929i)T^{2} \)
53 \( 1 + (0.621 - 0.783i)T^{2} \)
59 \( 1 + (0.378 - 1.62i)T + (-0.896 - 0.443i)T^{2} \)
61 \( 1 + (0.467 + 0.883i)T^{2} \)
67 \( 1 + (1.26 + 1.01i)T + (0.209 + 0.977i)T^{2} \)
71 \( 1 + (0.467 - 0.883i)T^{2} \)
73 \( 1 + (-1.22 + 0.786i)T + (0.418 - 0.908i)T^{2} \)
79 \( 1 + (0.227 + 0.973i)T^{2} \)
83 \( 1 + (1.27 - 0.600i)T + (0.635 - 0.771i)T^{2} \)
89 \( 1 + (-1.38 - 0.886i)T + (0.418 + 0.908i)T^{2} \)
97 \( 1 + (1.19 - 0.963i)T + (0.209 - 0.977i)T^{2} \)
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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

−9.398929406442447008953989642114, −8.290582281918226628191478316974, −7.76734340522975352273243822857, −6.86412057802517275774888833672, −6.01293077133157104640465872651, −5.57328384127498886324192158102, −4.69079511340595430750149690786, −4.05080415166705238945422800863, −3.31072356412010242097238018468, −1.80763623357186332105824432791, 0.78669034629824678665031479153, 1.59004423781613552368662665741, 3.07198585739786962154095532866, 3.57334894236757596841773304513, 4.84795135865930418959537087531, 5.49779100109176231445201462095, 6.15914894380360988559315581146, 6.84048029055037804672598672227, 7.57871775733713415503557523852, 8.799355614302862534183491727932

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