Properties

Label 2-2848-712.227-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.644 + 0.764i$
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  + (0.125 − 0.0683i)3-s + (−0.529 + 0.824i)9-s + (0.755 − 1.65i)11-s + (−0.281 − 0.0405i)17-s + (−0.398 − 1.83i)19-s + (0.654 − 0.755i)25-s + (−0.0201 + 0.281i)27-s + (−0.0185 − 0.258i)33-s + (0.677 − 1.24i)41-s + (0.559 + 1.50i)43-s + (0.755 + 0.654i)49-s + (−0.0380 + 0.0141i)51-s + (−0.175 − 0.202i)57-s + (1.05 + 0.574i)59-s + (−1.03 − 0.304i)67-s + ⋯
L(s)  = 1  + (0.125 − 0.0683i)3-s + (−0.529 + 0.824i)9-s + (0.755 − 1.65i)11-s + (−0.281 − 0.0405i)17-s + (−0.398 − 1.83i)19-s + (0.654 − 0.755i)25-s + (−0.0201 + 0.281i)27-s + (−0.0185 − 0.258i)33-s + (0.677 − 1.24i)41-s + (0.559 + 1.50i)43-s + (0.755 + 0.654i)49-s + (−0.0380 + 0.0141i)51-s + (−0.175 − 0.202i)57-s + (1.05 + 0.574i)59-s + (−1.03 − 0.304i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220402500\)
\(L(\frac12)\) \(\approx\) \(1.220402500\)
\(L(1)\) 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 \( 1 + (-0.654 - 0.755i)T \)
good3 \( 1 + (-0.125 + 0.0683i)T + (0.540 - 0.841i)T^{2} \)
5 \( 1 + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.755 - 0.654i)T^{2} \)
11 \( 1 + (-0.755 + 1.65i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.540 - 0.841i)T^{2} \)
17 \( 1 + (0.281 + 0.0405i)T + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.398 + 1.83i)T + (-0.909 + 0.415i)T^{2} \)
23 \( 1 + (-0.909 + 0.415i)T^{2} \)
29 \( 1 + (0.755 + 0.654i)T^{2} \)
31 \( 1 + (-0.909 - 0.415i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.677 + 1.24i)T + (-0.540 - 0.841i)T^{2} \)
43 \( 1 + (-0.559 - 1.50i)T + (-0.755 + 0.654i)T^{2} \)
47 \( 1 + (0.841 - 0.540i)T^{2} \)
53 \( 1 + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-1.05 - 0.574i)T + (0.540 + 0.841i)T^{2} \)
61 \( 1 + (-0.989 - 0.142i)T^{2} \)
67 \( 1 + (1.03 + 0.304i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (-1.27 + 0.817i)T + (0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (1.05 + 1.40i)T + (-0.281 + 0.959i)T^{2} \)
97 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
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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

−8.866605893313333016607193844534, −8.267729515683853386120542257506, −7.38451382889827242323293517261, −6.51063196725391876898327926521, −5.88402236423719804607664215917, −4.97645569138379863221731359609, −4.15369034826797928362831374417, −3.03694883662048760761816664071, −2.39220257628288744897424258708, −0.822203507543702557402628435883, 1.41469767536455374211800885159, 2.41776056518296880540702149586, 3.67055981628687678678140515661, 4.16118667293124393516745735628, 5.22718365045411067010126462881, 6.09973667391901481632832625613, 6.81389122392317672280775277990, 7.49183881670673726383668863546, 8.431813240298046572020935581616, 9.085767611940258477409791960958

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