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Properties

Label 2-1288-1288.1245-c0-0-1
Degree $2$
Conductor $1288$
Sign $0.381 + 0.924i$
Analytic cond. $0.642795$
Root an. cond. $0.801745$
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  + (0.142 + 0.989i)2-s + (−0.755 + 1.65i)3-s + (−0.959 + 0.281i)4-s + (−0.708 + 0.817i)5-s + (−1.74 − 0.512i)6-s + (−0.841 + 0.540i)7-s + (−0.415 − 0.909i)8-s + (−1.51 − 1.74i)9-s + (−0.909 − 0.584i)10-s + (0.258 − 1.80i)12-s + (1.66 + 1.07i)13-s + (−0.654 − 0.755i)14-s + (−0.817 − 1.78i)15-s + (0.841 − 0.540i)16-s + (1.51 − 1.74i)18-s + (−1.45 + 0.425i)19-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.755 + 1.65i)3-s + (−0.959 + 0.281i)4-s + (−0.708 + 0.817i)5-s + (−1.74 − 0.512i)6-s + (−0.841 + 0.540i)7-s + (−0.415 − 0.909i)8-s + (−1.51 − 1.74i)9-s + (−0.909 − 0.584i)10-s + (0.258 − 1.80i)12-s + (1.66 + 1.07i)13-s + (−0.654 − 0.755i)14-s + (−0.817 − 1.78i)15-s + (0.841 − 0.540i)16-s + (1.51 − 1.74i)18-s + (−1.45 + 0.425i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.381 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4493349967\)
\(L(\frac12)\) \(\approx\) \(0.4493349967\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.755 - 1.65i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (0.708 - 0.817i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-1.66 - 1.07i)T + (0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (1.27 + 0.817i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.368 + 0.425i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 + 0.989i)T^{2} \)
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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

−10.54711855718747017388886051528, −9.502923663343366676063567703981, −9.025755273403678870186647542350, −8.262474978916778757366625175345, −6.82528461330251995363697986624, −6.32279018472022414252823900513, −5.65614435092730609505542218772, −4.54500546198496541210163218400, −3.78406612524401450635149966878, −3.30552019286249405983682800955, 0.45960225095452461951426694721, 1.28461879307519293115988975075, 2.72449750722975983847154420679, 3.85306246994161733935448524973, 4.90351686939739076635649378136, 5.97694123631411251075489352296, 6.53396539613971461480327373233, 7.70787505501362026909828852621, 8.409644202292727177353730256796, 8.980788024712778879719340943332

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