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Properties

Label 2-2872-2872.717-c0-0-0
Degree $2$
Conductor $2872$
Sign $0.245 + 0.969i$
Analytic cond. $1.43331$
Root an. cond. $1.19721$
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  + (0.0825 + 0.996i)2-s + 1.83i·3-s + (−0.986 + 0.164i)4-s − 1.22i·5-s + (−1.82 + 0.151i)6-s + (−0.245 − 0.969i)8-s − 2.35·9-s + (1.22 − 0.101i)10-s + 1.67i·11-s + (−0.301 − 1.80i)12-s + 2.24·15-s + (0.945 − 0.324i)16-s − 1.89·17-s + (−0.194 − 2.34i)18-s + (0.202 + 1.21i)20-s + ⋯
L(s)  = 1  + (0.0825 + 0.996i)2-s + 1.83i·3-s + (−0.986 + 0.164i)4-s − 1.22i·5-s + (−1.82 + 0.151i)6-s + (−0.245 − 0.969i)8-s − 2.35·9-s + (1.22 − 0.101i)10-s + 1.67i·11-s + (−0.301 − 1.80i)12-s + 2.24·15-s + (0.945 − 0.324i)16-s − 1.89·17-s + (−0.194 − 2.34i)18-s + (0.202 + 1.21i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(1.43331\)
Root analytic conductor: \(1.19721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2872} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2872,\ (\ :0),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3334325255\)
\(L(\frac12)\) \(\approx\) \(0.3334325255\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0825 - 0.996i)T \)
359 \( 1 + T \)
good3 \( 1 - 1.83iT - T^{2} \)
5 \( 1 + 1.22iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.67iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.89T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.97T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.951iT - T^{2} \)
41 \( 1 + 1.75T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.35T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.165T + T^{2} \)
79 \( 1 + 1.57T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.313772297289190971665189681727, −8.905525276723866157189287131455, −8.337763326897112999902076148927, −7.35479680695918447107690431202, −6.32568174794160240025905916905, −5.42671761754782235892641665369, −4.83949434633455457552172836308, −4.24235330094011531511321600930, −3.93419597936410016543128787520, −2.21951451281488399377023570658, 0.19255904285134469563879761273, 1.65194485585104052133900903630, 2.48732832536541494124530499675, 3.05857625181260416709182052322, 4.05509202372573375120158728084, 5.56629993405332843838387670656, 6.21601621295233142436910835723, 6.73367350906869625892833245690, 7.64954750311762435670327712229, 8.518728994505978530529301663648

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