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Properties

Label 2-2888-2888.2875-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.175 + 0.984i$
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  + (−0.119 − 0.992i)2-s + (−0.802 − 0.648i)3-s + (−0.971 + 0.236i)4-s + (−0.548 + 0.873i)6-s + (0.350 + 0.936i)8-s + (0.0134 + 0.0628i)9-s + (−0.119 − 0.0483i)11-s + (0.932 + 0.440i)12-s + (0.888 − 0.459i)16-s + (0.548 + 1.87i)17-s + (0.0608 − 0.0208i)18-s + (0.919 − 0.393i)19-s + (−0.0338 + 0.123i)22-s + (0.325 − 0.978i)24-s + (0.418 + 0.908i)25-s + ⋯
L(s)  = 1  + (−0.119 − 0.992i)2-s + (−0.802 − 0.648i)3-s + (−0.971 + 0.236i)4-s + (−0.548 + 0.873i)6-s + (0.350 + 0.936i)8-s + (0.0134 + 0.0628i)9-s + (−0.119 − 0.0483i)11-s + (0.932 + 0.440i)12-s + (0.888 − 0.459i)16-s + (0.548 + 1.87i)17-s + (0.0608 − 0.0208i)18-s + (0.919 − 0.393i)19-s + (−0.0338 + 0.123i)22-s + (0.325 − 0.978i)24-s + (0.418 + 0.908i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2875, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.175 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7701547764\)
\(L(\frac12)\) \(\approx\) \(0.7701547764\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.119 + 0.992i)T \)
19 \( 1 + (-0.919 + 0.393i)T \)
good3 \( 1 + (0.802 + 0.648i)T + (0.209 + 0.977i)T^{2} \)
5 \( 1 + (-0.418 - 0.908i)T^{2} \)
7 \( 1 + (0.298 - 0.954i)T^{2} \)
11 \( 1 + (0.119 + 0.0483i)T + (0.716 + 0.697i)T^{2} \)
13 \( 1 + (-0.999 + 0.0367i)T^{2} \)
17 \( 1 + (-0.548 - 1.87i)T + (-0.842 + 0.539i)T^{2} \)
23 \( 1 + (0.951 - 0.307i)T^{2} \)
29 \( 1 + (-0.606 + 0.794i)T^{2} \)
31 \( 1 + (-0.993 + 0.110i)T^{2} \)
37 \( 1 + (-0.245 + 0.969i)T^{2} \)
41 \( 1 + (-0.957 + 0.141i)T + (0.957 - 0.289i)T^{2} \)
43 \( 1 + (-0.900 - 0.0663i)T + (0.989 + 0.146i)T^{2} \)
47 \( 1 + (-0.997 - 0.0734i)T^{2} \)
53 \( 1 + (0.562 - 0.826i)T^{2} \)
59 \( 1 + (-0.167 + 0.423i)T + (-0.729 - 0.684i)T^{2} \)
61 \( 1 + (0.999 + 0.0183i)T^{2} \)
67 \( 1 + (0.964 + 1.12i)T + (-0.155 + 0.987i)T^{2} \)
71 \( 1 + (0.999 - 0.0183i)T^{2} \)
73 \( 1 + (-1.36 + 1.43i)T + (-0.0459 - 0.998i)T^{2} \)
79 \( 1 + (0.367 + 0.929i)T^{2} \)
83 \( 1 + (-1.18 + 0.265i)T + (0.904 - 0.426i)T^{2} \)
89 \( 1 + (0.412 + 0.431i)T + (-0.0459 + 0.998i)T^{2} \)
97 \( 1 + (0.225 - 0.263i)T + (-0.155 - 0.987i)T^{2} \)
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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

−9.027374508808773158215160250715, −7.977085678699900966975749517823, −7.46030251515618561441904184388, −6.33139931290281771455271228122, −5.70601151027045167819960183254, −4.92088636429344325295801655605, −3.84103100645856704402632507082, −3.10196091809532348033417730788, −1.81150222354803876131550989112, −0.959295076268416423177681494075, 0.801807097694089943444399447757, 2.72054435184244681306082778481, 3.93756387093649311457288594275, 4.77466338976334304172901059717, 5.32356821301574100754364992626, 5.90072950279565110062038307649, 6.87527207995595145251661892386, 7.52923444382840321849253381860, 8.224175471102038830940426064164, 9.208347857200980236010512204664

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