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Properties

Label 2-2888-152.99-c0-0-2
Degree $2$
Conductor $2888$
Sign $-0.291 - 0.956i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.500 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)16-s + (−1.87 − 0.684i)17-s + (0.173 + 0.984i)22-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)27-s + (−0.173 + 0.984i)32-s + (−0.766 + 0.642i)33-s + (−1.53 − 1.28i)34-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.500 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)16-s + (−1.87 − 0.684i)17-s + (0.173 + 0.984i)22-s + (−0.766 + 0.642i)24-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)27-s + (−0.173 + 0.984i)32-s + (−0.766 + 0.642i)33-s + (−1.53 − 1.28i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.291 - 0.956i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.291 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.405738723\)
\(L(\frac12)\) \(\approx\) \(2.405738723\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.142614725482760507621219798773, −8.516947318639450949164803406048, −7.40422277572683460040562400483, −6.82594328700763310578477546074, −6.12423819253308763202825705296, −4.96048075931873541715188645971, −4.48165354001040694099559989639, −3.97161662395227114114006426602, −2.88975911218576581705740135183, −1.97008531857657750542271789287, 1.18755028154528213202417086321, 2.07093508508354573964051111952, 2.95897381007437452914083944871, 4.01037405205281024457426025419, 4.66916094282393279337459524968, 5.87809327423722959923681646222, 6.33414739980869147114525093714, 7.06383271206954371160416004651, 7.73042402357062780146667694516, 8.726459383623654953293001012403

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