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Properties

Label 2-2888-2888.35-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.586 - 0.809i$
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  + (0.933 + 0.359i)2-s + (−1.23 − 1.29i)3-s + (0.741 + 0.670i)4-s + (−0.689 − 1.65i)6-s + (0.451 + 0.892i)8-s + (−0.101 + 2.21i)9-s + (−1.03 + 0.232i)11-s + (−0.0493 − 1.79i)12-s + (0.100 + 0.994i)16-s + (0.300 + 1.40i)17-s + (−0.889 + 2.02i)18-s + (−0.155 + 0.987i)19-s + (−1.05 − 0.155i)22-s + (0.597 − 1.69i)24-s + (0.663 + 0.748i)25-s + ⋯
L(s)  = 1  + (0.933 + 0.359i)2-s + (−1.23 − 1.29i)3-s + (0.741 + 0.670i)4-s + (−0.689 − 1.65i)6-s + (0.451 + 0.892i)8-s + (−0.101 + 2.21i)9-s + (−1.03 + 0.232i)11-s + (−0.0493 − 1.79i)12-s + (0.100 + 0.994i)16-s + (0.300 + 1.40i)17-s + (−0.889 + 2.02i)18-s + (−0.155 + 0.987i)19-s + (−1.05 − 0.155i)22-s + (0.597 − 1.69i)24-s + (0.663 + 0.748i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.586 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218211982\)
\(L(\frac12)\) \(\approx\) \(1.218211982\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.933 - 0.359i)T \)
19 \( 1 + (0.155 - 0.987i)T \)
good3 \( 1 + (1.23 + 1.29i)T + (-0.0459 + 0.998i)T^{2} \)
5 \( 1 + (-0.663 - 0.748i)T^{2} \)
7 \( 1 + (0.962 + 0.272i)T^{2} \)
11 \( 1 + (1.03 - 0.232i)T + (0.904 - 0.426i)T^{2} \)
13 \( 1 + (0.991 - 0.128i)T^{2} \)
17 \( 1 + (-0.300 - 1.40i)T + (-0.912 + 0.410i)T^{2} \)
23 \( 1 + (-0.888 - 0.459i)T^{2} \)
29 \( 1 + (-0.997 + 0.0734i)T^{2} \)
31 \( 1 + (0.926 - 0.376i)T^{2} \)
37 \( 1 + (0.0825 - 0.996i)T^{2} \)
41 \( 1 + (-1.44 + 0.818i)T + (0.515 - 0.856i)T^{2} \)
43 \( 1 + (1.45 + 0.383i)T + (0.870 + 0.492i)T^{2} \)
47 \( 1 + (-0.967 - 0.254i)T^{2} \)
53 \( 1 + (0.263 + 0.964i)T^{2} \)
59 \( 1 + (-1.72 - 1.01i)T + (0.484 + 0.875i)T^{2} \)
61 \( 1 + (-0.0642 + 0.997i)T^{2} \)
67 \( 1 + (-0.200 - 1.67i)T + (-0.971 + 0.236i)T^{2} \)
71 \( 1 + (-0.0642 - 0.997i)T^{2} \)
73 \( 1 + (1.65 + 0.535i)T + (0.811 + 0.584i)T^{2} \)
79 \( 1 + (0.861 - 0.507i)T^{2} \)
83 \( 1 + (-1.42 + 1.39i)T + (0.0275 - 0.999i)T^{2} \)
89 \( 1 + (-1.83 + 0.591i)T + (0.811 - 0.584i)T^{2} \)
97 \( 1 + (0.0413 - 0.344i)T + (-0.971 - 0.236i)T^{2} \)
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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

−8.596729586914462283210722208957, −7.902647166462993238123914289129, −7.36922302590277590031581992546, −6.67012347123771251754421929212, −5.87993700026142360895408502006, −5.52390526663170564764544116201, −4.70220865858619184575460754194, −3.58817058234119887064734332336, −2.33574414062130736570322496216, −1.49537276080373075249705821697, 0.66062635653877606257787282804, 2.60719190648726731473138080497, 3.33844427227054953715732191618, 4.42395332316576858638596785035, 4.97369992980009371277099699279, 5.35752108379103893314657576251, 6.31188307787925147839273817429, 6.88507884978425198222913216852, 8.021887230645700837065960957685, 9.338759876195983356315350622376

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