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Properties

Label 2-928-116.99-c1-0-9
Degree $2$
Conductor $928$
Sign $0.451 - 0.892i$
Analytic cond. $7.41011$
Root an. cond. $2.72215$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.01 − 1.01i)3-s + 3.11i·5-s − 4.60i·7-s + 0.932i·9-s + (−4.52 + 4.52i)11-s + 4.86i·13-s + (3.16 + 3.16i)15-s + (4.26 + 4.26i)17-s + (−1.06 + 1.06i)19-s + (−4.68 − 4.68i)21-s + 1.42i·23-s − 4.68·25-s + (3.99 + 3.99i)27-s + (3.29 − 4.26i)29-s + (3.27 − 3.27i)31-s + ⋯
L(s)  = 1  + (0.587 − 0.587i)3-s + 1.39i·5-s − 1.74i·7-s + 0.310i·9-s + (−1.36 + 1.36i)11-s + 1.34i·13-s + (0.817 + 0.817i)15-s + (1.03 + 1.03i)17-s + (−0.245 + 0.245i)19-s + (−1.02 − 1.02i)21-s + 0.297i·23-s − 0.937·25-s + (0.769 + 0.769i)27-s + (0.611 − 0.791i)29-s + (0.588 − 0.588i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(928\)    =    \(2^{5} \cdot 29\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(7.41011\)
Root analytic conductor: \(2.72215\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{928} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 928,\ (\ :1/2),\ 0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38927 + 0.854532i\)
\(L(\frac12)\) \(\approx\) \(1.38927 + 0.854532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-3.29 + 4.26i)T \)
good3 \( 1 + (-1.01 + 1.01i)T - 3iT^{2} \)
5 \( 1 - 3.11iT - 5T^{2} \)
7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 + (4.52 - 4.52i)T - 11iT^{2} \)
13 \( 1 - 4.86iT - 13T^{2} \)
17 \( 1 + (-4.26 - 4.26i)T + 17iT^{2} \)
19 \( 1 + (1.06 - 1.06i)T - 19iT^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
31 \( 1 + (-3.27 + 3.27i)T - 31iT^{2} \)
37 \( 1 + (-2.35 + 2.35i)T - 37iT^{2} \)
41 \( 1 + (-0.852 + 0.852i)T - 41iT^{2} \)
43 \( 1 + (3.66 - 3.66i)T - 43iT^{2} \)
47 \( 1 + (1.16 + 1.16i)T + 47iT^{2} \)
53 \( 1 - 1.71T + 53T^{2} \)
59 \( 1 - 8.79iT - 59T^{2} \)
61 \( 1 + (4.92 + 4.92i)T + 61iT^{2} \)
67 \( 1 - 0.363T + 67T^{2} \)
71 \( 1 - 0.191T + 71T^{2} \)
73 \( 1 + (-1.34 + 1.34i)T - 73iT^{2} \)
79 \( 1 + (2.34 - 2.34i)T - 79iT^{2} \)
83 \( 1 + 9.58iT - 83T^{2} \)
89 \( 1 + (-12.9 - 12.9i)T + 89iT^{2} \)
97 \( 1 + (-9.78 + 9.78i)T - 97iT^{2} \)
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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

−10.37139160703146159313847847415, −9.702700054160374481931850570158, −8.052650920622266817797900321150, −7.60291250126121119277309287147, −7.08405599840996700862182334914, −6.30017018899629469061484739089, −4.69756575237260181436728665521, −3.82846320862840185443850179648, −2.66028748677331495028874287228, −1.71569221235302237151361054815, 0.72051991118483072556951586459, 2.76050720571887974862317092408, 3.19733479584874601786280856430, 4.98468901714193786454515712871, 5.27272744886577219310429830572, 6.11971035156340680902921016015, 7.952993113319425180848089185484, 8.467171126426185847874492565309, 8.884569500785643422439943241431, 9.721331665669725472929258862670

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