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Properties

Label 2-2888-2888.2443-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.171 - 0.985i$
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  + (0.451 + 0.892i)2-s + (0.289 − 0.251i)3-s + (−0.592 + 0.805i)4-s + (0.355 + 0.144i)6-s + (−0.986 − 0.164i)8-s + (−0.117 + 0.844i)9-s + (1.56 − 1.22i)11-s + (0.0316 + 0.382i)12-s + (−0.298 − 0.954i)16-s + (0.802 + 1.09i)17-s + (−0.806 + 0.276i)18-s + (0.451 − 0.892i)19-s + (1.79 + 0.848i)22-s + (−0.326 + 0.200i)24-s + (−0.821 + 0.569i)25-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + (0.289 − 0.251i)3-s + (−0.592 + 0.805i)4-s + (0.355 + 0.144i)6-s + (−0.986 − 0.164i)8-s + (−0.117 + 0.844i)9-s + (1.56 − 1.22i)11-s + (0.0316 + 0.382i)12-s + (−0.298 − 0.954i)16-s + (0.802 + 1.09i)17-s + (−0.806 + 0.276i)18-s + (0.451 − 0.892i)19-s + (1.79 + 0.848i)22-s + (−0.326 + 0.200i)24-s + (−0.821 + 0.569i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.171 - 0.985i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.171 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722122344\)
\(L(\frac12)\) \(\approx\) \(1.722122344\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.451 - 0.892i)T \)
19 \( 1 + (-0.451 + 0.892i)T \)
good3 \( 1 + (-0.289 + 0.251i)T + (0.137 - 0.990i)T^{2} \)
5 \( 1 + (0.821 - 0.569i)T^{2} \)
7 \( 1 + (0.677 + 0.735i)T^{2} \)
11 \( 1 + (-1.56 + 1.22i)T + (0.245 - 0.969i)T^{2} \)
13 \( 1 + (0.926 - 0.376i)T^{2} \)
17 \( 1 + (-0.802 - 1.09i)T + (-0.298 + 0.954i)T^{2} \)
23 \( 1 + (-0.137 - 0.990i)T^{2} \)
29 \( 1 + (-0.975 + 0.218i)T^{2} \)
31 \( 1 + (0.401 - 0.915i)T^{2} \)
37 \( 1 + (-0.245 + 0.969i)T^{2} \)
41 \( 1 + (0.0105 - 0.383i)T + (-0.998 - 0.0550i)T^{2} \)
43 \( 1 + (-0.784 - 0.762i)T + (0.0275 + 0.999i)T^{2} \)
47 \( 1 + (-0.716 - 0.697i)T^{2} \)
53 \( 1 + (-0.716 - 0.697i)T^{2} \)
59 \( 1 + (0.0550 - 1.99i)T + (-0.998 - 0.0550i)T^{2} \)
61 \( 1 + (0.191 - 0.981i)T^{2} \)
67 \( 1 + (-0.0963 - 0.257i)T + (-0.754 + 0.656i)T^{2} \)
71 \( 1 + (0.191 + 0.981i)T^{2} \)
73 \( 1 + (0.0326 + 0.0444i)T + (-0.298 + 0.954i)T^{2} \)
79 \( 1 + (-0.0275 - 0.999i)T^{2} \)
83 \( 1 + (1.32 + 1.43i)T + (-0.0825 + 0.996i)T^{2} \)
89 \( 1 + (-0.802 + 1.09i)T + (-0.298 - 0.954i)T^{2} \)
97 \( 1 + (0.350 - 0.936i)T + (-0.754 - 0.656i)T^{2} \)
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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

−8.884481065322751509745225484222, −8.245714181892702899889502018754, −7.61861555190709177090272482935, −6.83495185479699703232918567731, −6.01849114288980590336278265217, −5.52155460213278974610348231077, −4.43645293825392212524326625360, −3.65040010204469203748428317126, −2.88521120348346924512182232783, −1.37577058916253336084178249080, 1.11056831339458464043510728864, 2.14372258851521452192732828042, 3.30093322820434636775392600195, 3.89801356248143046975433249919, 4.59072150292019308769696338159, 5.59067257234575413655119643095, 6.38915340773143038965425420212, 7.16617528377157587309925405381, 8.232239021406217596319063405370, 9.196580415610285507842113702221

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