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Properties

Label 2-2912-728.181-c0-0-3
Degree $2$
Conductor $2912$
Sign $0.707 - 0.707i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + 1.41·3-s + 1.41i·5-s − i·7-s + 1.00·9-s + (−0.707 + 0.707i)13-s + 2.00i·15-s + 1.41i·19-s − 1.41i·21-s + 2·23-s − 1.00·25-s + 1.41·35-s + (−1.00 + 1.00i)39-s + 1.41i·45-s − 49-s + 2.00i·57-s + ⋯
L(s)  = 1  + 1.41·3-s + 1.41i·5-s − i·7-s + 1.00·9-s + (−0.707 + 0.707i)13-s + 2.00i·15-s + 1.41i·19-s − 1.41i·21-s + 2·23-s − 1.00·25-s + 1.41·35-s + (−1.00 + 1.00i)39-s + 1.41i·45-s − 49-s + 2.00i·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (2001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.960769330\)
\(L(\frac12)\) \(\approx\) \(1.960769330\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
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

−9.096853250913227653152041428495, −8.134641940371039077383835174295, −7.51262202172533071485497857967, −7.00814237672911695534229974041, −6.36572528731631120153281003542, −4.99963572458858540318388140270, −3.92916743748449040022177749409, −3.36273230211768659078341144150, −2.65681194409379412654779901492, −1.68251305297090407784206146699, 1.14267381849883199594566833199, 2.49101923101141950941641141133, 2.88441013950534007533070264167, 4.10721004759174667684788453529, 5.09557412397718818382599342686, 5.37920740541460559824249463172, 6.80036122199674012768306890463, 7.58806586911730797748556198484, 8.449985650385217894431702086506, 8.770579520087997646255370463763

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