Properties

Label 2-2816-176.109-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(1.016525018\)
\(L(\frac12)\) \(\approx\) \(1.016525018\)
\(L(1)\) 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

−8.661267137293001770743882476635, −8.353961211102728380917208693115, −7.28572984054204261553904440524, −6.57792160441591041685567093946, −6.26944200484621058482243916145, −4.93923332555102522118589585438, −3.88766111762336616136531192747, −3.45817856747622612695722570772, −2.51953187921625064160581457083, −0.789377591935566481869410101352, 1.16224909733994835325452900759, 2.29308011262281494580008967787, 3.66494610432392043348271454496, 4.33122205139005145221947986412, 5.01692769138541073303605143716, 5.78760211386250580322615304492, 7.04940085311661970380943351403, 7.53094654587579524726020090060, 8.303836688579101861185230632212, 8.872647673728486559628780413239

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