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Properties

Label 2-2888-2888.2475-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.973 + 0.227i$
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  + (0.100 + 0.994i)2-s + (0.524 + 0.0482i)3-s + (−0.979 + 0.200i)4-s + (0.00483 + 0.526i)6-s + (−0.298 − 0.954i)8-s + (−0.710 − 0.132i)9-s + (−0.790 + 0.959i)11-s + (−0.523 + 0.0579i)12-s + (0.919 − 0.393i)16-s + (−1.32 + 1.49i)17-s + (0.0596 − 0.720i)18-s + (0.811 + 0.584i)19-s + (−1.03 − 0.689i)22-s + (−0.110 − 0.514i)24-s + (−0.971 − 0.236i)25-s + ⋯
L(s)  = 1  + (0.100 + 0.994i)2-s + (0.524 + 0.0482i)3-s + (−0.979 + 0.200i)4-s + (0.00483 + 0.526i)6-s + (−0.298 − 0.954i)8-s + (−0.710 − 0.132i)9-s + (−0.790 + 0.959i)11-s + (−0.523 + 0.0579i)12-s + (0.919 − 0.393i)16-s + (−1.32 + 1.49i)17-s + (0.0596 − 0.720i)18-s + (0.811 + 0.584i)19-s + (−1.03 − 0.689i)22-s + (−0.110 − 0.514i)24-s + (−0.971 − 0.236i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.973 + 0.227i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.973 + 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7054000451\)
\(L(\frac12)\) \(\approx\) \(0.7054000451\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.100 - 0.994i)T \)
19 \( 1 + (-0.811 - 0.584i)T \)
good3 \( 1 + (-0.524 - 0.0482i)T + (0.983 + 0.182i)T^{2} \)
5 \( 1 + (0.971 + 0.236i)T^{2} \)
7 \( 1 + (-0.451 - 0.892i)T^{2} \)
11 \( 1 + (0.790 - 0.959i)T + (-0.191 - 0.981i)T^{2} \)
13 \( 1 + (-0.870 + 0.492i)T^{2} \)
17 \( 1 + (1.32 - 1.49i)T + (-0.119 - 0.992i)T^{2} \)
23 \( 1 + (0.333 - 0.942i)T^{2} \)
29 \( 1 + (-0.957 + 0.289i)T^{2} \)
31 \( 1 + (-0.0275 + 0.999i)T^{2} \)
37 \( 1 + (-0.945 - 0.324i)T^{2} \)
41 \( 1 + (-0.658 - 1.24i)T + (-0.562 + 0.826i)T^{2} \)
43 \( 1 + (0.992 + 1.64i)T + (-0.467 + 0.883i)T^{2} \)
47 \( 1 + (-0.515 - 0.856i)T^{2} \)
53 \( 1 + (-0.484 + 0.875i)T^{2} \)
59 \( 1 + (1.06 - 1.68i)T + (-0.435 - 0.900i)T^{2} \)
61 \( 1 + (-0.967 - 0.254i)T^{2} \)
67 \( 1 + (1.15 - 0.597i)T + (0.577 - 0.816i)T^{2} \)
71 \( 1 + (-0.967 + 0.254i)T^{2} \)
73 \( 1 + (0.295 + 0.887i)T + (-0.800 + 0.599i)T^{2} \)
79 \( 1 + (0.531 + 0.847i)T^{2} \)
83 \( 1 + (1.91 + 0.105i)T + (0.993 + 0.110i)T^{2} \)
89 \( 1 + (-0.285 + 0.856i)T + (-0.800 - 0.599i)T^{2} \)
97 \( 1 + (-1.36 - 0.704i)T + (0.577 + 0.816i)T^{2} \)
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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

−9.009296935246551411545304625798, −8.568069718913013682492802534009, −7.76195905378011029837100297275, −7.31408595073230961988858257667, −6.16662375367260282239516002722, −5.77812022958436038363722414281, −4.69142091370800238804882154766, −4.02704655589065224687927219827, −3.04825020238291820807365152885, −1.90150840085912928240152871295, 0.37354813662508798183450262918, 2.04788868906967215980587851214, 2.85482157181388104971718821459, 3.35943940341733166878769025578, 4.56788801875314677086375165620, 5.27524321620059086357146368500, 6.00967694450430605897581482791, 7.25923350740488042989817990162, 8.050120275587419291885577668627, 8.734020544677144871378635952965

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