Properties

Label 2-2816-176.109-c0-0-0
Degree $2$
Conductor $2816$
Sign $-0.923 - 0.382i$
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 + i)5-s − i·9-s + (−0.707 + 0.707i)11-s + 1.41i·23-s − i·25-s − 1.41·31-s + (−1 + i)37-s + (1 + i)45-s − 1.41·47-s − 49-s + (1 − i)53-s − 1.41i·55-s + (−1.41 + 1.41i)59-s + (−1.41 − 1.41i)67-s + 1.41i·71-s + ⋯
L(s)  = 1  + (−1 + i)5-s − i·9-s + (−0.707 + 0.707i)11-s + 1.41i·23-s − i·25-s − 1.41·31-s + (−1 + i)37-s + (1 + i)45-s − 1.41·47-s − 49-s + (1 − i)53-s − 1.41i·55-s + (−1.41 + 1.41i)59-s + (−1.41 − 1.41i)67-s + 1.41i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 2816,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3858317006\)
\(L(\frac12)\) \(\approx\) \(0.3858317006\)
\(L(1)\) 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 \( 1 + iT^{2} \)
5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−9.371183746135803350292055444163, −8.466893260667531216609011028575, −7.56950740771067360544440283742, −7.19899560172908985605952652517, −6.44729990260741441673877109822, −5.48495240447859968329597211637, −4.51501097906894067326978060524, −3.49248000699343118583012223489, −3.14948082130405839723824891438, −1.74709495479259312714727736905, 0.23499735478168037607988433200, 1.79613614095432532947097272571, 2.98631757583797241437073466133, 3.97457079391298244552890689623, 4.82356542827937433082752603940, 5.29915923065454176558660077318, 6.31910129370731637584216949439, 7.48327717992847370847291806652, 7.88654352517612807365504591693, 8.620157425108492989074810372120

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