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Properties

Label 2-2816-44.3-c0-0-3
Degree $2$
Conductor $2816$
Sign $0.794 + 0.606i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
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  + (1.53 − 0.5i)3-s + (1.30 − 0.951i)9-s + (0.951 − 0.309i)11-s + (−0.5 − 0.363i)17-s + (−0.587 + 0.190i)19-s + (−0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (1.30 − 0.951i)33-s + (0.5 + 1.53i)41-s + 1.61i·43-s + (−0.809 − 0.587i)49-s + (−0.951 − 0.309i)51-s + (−0.809 + 0.587i)57-s + (−0.587 − 0.190i)59-s + 0.618i·67-s + ⋯
L(s)  = 1  + (1.53 − 0.5i)3-s + (1.30 − 0.951i)9-s + (0.951 − 0.309i)11-s + (−0.5 − 0.363i)17-s + (−0.587 + 0.190i)19-s + (−0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (1.30 − 0.951i)33-s + (0.5 + 1.53i)41-s + 1.61i·43-s + (−0.809 − 0.587i)49-s + (−0.951 − 0.309i)51-s + (−0.809 + 0.587i)57-s + (−0.587 − 0.190i)59-s + 0.618i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.134406301\)
\(L(\frac12)\) \(\approx\) \(2.134406301\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 - 0.618iT - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.846434230063834507584207944203, −8.144309539882983515407101578353, −7.65984441499949598524824227209, −6.61798966994319115582224656921, −6.23230361277029183567431020637, −4.73958561420993123285677413583, −3.96715278864409276170979523774, −3.11994574569179820707361655410, −2.31588812101157291477575477097, −1.34172929341801612404786277837, 1.69960623488584983869665353573, 2.47663114960204120496842655053, 3.59881277345529989745091643937, 4.00782481824576818780565372848, 4.92241717424490618205257259958, 6.07827477411466454283656811665, 7.02041701532578391532087692675, 7.64509451103101118507603062091, 8.560729154697439323088919609850, 8.999273964837431119153165922287

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