Properties

Label 2-2816-176.109-c0-0-8
Degree $2$
Conductor $2816$
Sign $0.793 - 0.608i$
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  + (1.22 + 1.22i)3-s + (1.36 − 1.36i)5-s + 1.99i·9-s + (−0.707 + 0.707i)11-s + 3.34·15-s + 0.517i·23-s − 2.73i·25-s + (−1.22 + 1.22i)27-s − 0.517·31-s − 1.73·33-s + (−0.366 + 0.366i)37-s + (2.73 + 2.73i)45-s − 1.41·47-s − 49-s + (1 − i)53-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)3-s + (1.36 − 1.36i)5-s + 1.99i·9-s + (−0.707 + 0.707i)11-s + 3.34·15-s + 0.517i·23-s − 2.73i·25-s + (−1.22 + 1.22i)27-s − 0.517·31-s − 1.73·33-s + (−0.366 + 0.366i)37-s + (2.73 + 2.73i)45-s − 1.41·47-s − 49-s + (1 − i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :0),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.234004675\)
\(L(\frac12)\) \(\approx\) \(2.234004675\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
5 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - 0.517iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + 0.517T + T^{2} \)
37 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.343649301539579413096638437401, −8.344395063382942602200973015035, −8.109240893253042547952983899194, −6.80372414731183350682765032147, −5.57076210511188607342275094752, −5.05001483928439612685288122735, −4.48462565126190499000784037904, −3.46930334568960363671694593152, −2.39395593674862896172022368423, −1.70465992718193660575128918989, 1.49214066517129828031480848210, 2.39184140279344478798261153826, 2.85063599658913618024452283641, 3.63865794396719499897069828100, 5.35340127982904766113385199307, 6.11610706394733439677097249384, 6.75367000158910726365046883970, 7.30524596332136832799469767045, 8.090688694713523007395640300011, 8.809703445719548429189775031746

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