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Properties

Label 2-2848-712.347-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.656 - 0.754i$
Analytic cond. $1.42133$
Root an. cond. $1.19219$
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  + (−0.0303 + 0.424i)3-s + (0.810 + 0.116i)9-s + (−0.540 + 0.158i)11-s + (0.755 + 0.345i)17-s + (0.574 + 0.767i)19-s + (−0.841 − 0.540i)25-s + (−0.164 + 0.755i)27-s + (−0.0509 − 0.234i)33-s + (1.41 − 0.100i)41-s + (0.613 + 0.334i)43-s + (−0.540 + 0.841i)49-s + (−0.169 + 0.310i)51-s + (−0.342 + 0.220i)57-s + (−0.133 − 1.86i)59-s + (1.29 + 1.49i)67-s + ⋯
L(s)  = 1  + (−0.0303 + 0.424i)3-s + (0.810 + 0.116i)9-s + (−0.540 + 0.158i)11-s + (0.755 + 0.345i)17-s + (0.574 + 0.767i)19-s + (−0.841 − 0.540i)25-s + (−0.164 + 0.755i)27-s + (−0.0509 − 0.234i)33-s + (1.41 − 0.100i)41-s + (0.613 + 0.334i)43-s + (−0.540 + 0.841i)49-s + (−0.169 + 0.310i)51-s + (−0.342 + 0.220i)57-s + (−0.133 − 1.86i)59-s + (1.29 + 1.49i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (2127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286261036\)
\(L(\frac12)\) \(\approx\) \(1.286261036\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (0.0303 - 0.424i)T + (-0.989 - 0.142i)T^{2} \)
5 \( 1 + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.540 - 0.841i)T^{2} \)
11 \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.989 - 0.142i)T^{2} \)
17 \( 1 + (-0.755 - 0.345i)T + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.574 - 0.767i)T + (-0.281 + 0.959i)T^{2} \)
23 \( 1 + (-0.281 + 0.959i)T^{2} \)
29 \( 1 + (-0.540 + 0.841i)T^{2} \)
31 \( 1 + (-0.281 - 0.959i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1.41 + 0.100i)T + (0.989 - 0.142i)T^{2} \)
43 \( 1 + (-0.613 - 0.334i)T + (0.540 + 0.841i)T^{2} \)
47 \( 1 + (-0.142 - 0.989i)T^{2} \)
53 \( 1 + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.133 + 1.86i)T + (-0.989 + 0.142i)T^{2} \)
61 \( 1 + (-0.909 - 0.415i)T^{2} \)
67 \( 1 + (-1.29 - 1.49i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.153 - 1.07i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.133 + 0.0498i)T + (0.755 - 0.654i)T^{2} \)
97 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
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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

−9.304858551701041774602698335877, −8.061794880320154654593408416988, −7.77781760965464943432148078161, −6.84253602437588749595118607894, −5.88296198406468655188789696300, −5.22532354685694816135687259936, −4.27852596445519604645820147766, −3.63915291493960574320891619911, −2.50074552459976292686972253787, −1.33323483779851242300510945850, 0.950922973452269711800928025160, 2.14117434768623964434209838035, 3.16015574989485691897671646293, 4.11963700763235488774788235859, 5.05746576868859268799630843169, 5.80060952223044005565097467698, 6.68284365506631672477749660568, 7.54284617742661470467988194714, 7.77538986932831061315764858812, 8.934086898693578217985251562900

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