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Properties

Label 2-928-116.99-c1-0-26
Degree $2$
Conductor $928$
Sign $-0.401 + 0.915i$
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  + (2.19 − 2.19i)3-s − 0.390i·5-s − 0.923i·7-s − 6.59i·9-s + (0.993 − 0.993i)11-s − 0.641i·13-s + (−0.854 − 0.854i)15-s + (−4.43 − 4.43i)17-s + (−1.72 + 1.72i)19-s + (−2.02 − 2.02i)21-s + 7.45i·23-s + 4.84·25-s + (−7.87 − 7.87i)27-s + (−3.05 + 4.43i)29-s + (2.62 − 2.62i)31-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s − 0.174i·5-s − 0.348i·7-s − 2.19i·9-s + (0.299 − 0.299i)11-s − 0.177i·13-s + (−0.220 − 0.220i)15-s + (−1.07 − 1.07i)17-s + (−0.395 + 0.395i)19-s + (−0.441 − 0.441i)21-s + 1.55i·23-s + 0.969·25-s + (−1.51 − 1.51i)27-s + (−0.567 + 0.823i)29-s + (0.471 − 0.471i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(928\)    =    \(2^{5} \cdot 29\)
Sign: $-0.401 + 0.915i$
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: \((2,\ 928,\ (\ :1/2),\ -0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25837 - 1.92491i\)
\(L(\frac12)\) \(\approx\) \(1.25837 - 1.92491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.05 - 4.43i)T \)
good3 \( 1 + (-2.19 + 2.19i)T - 3iT^{2} \)
5 \( 1 + 0.390iT - 5T^{2} \)
7 \( 1 + 0.923iT - 7T^{2} \)
11 \( 1 + (-0.993 + 0.993i)T - 11iT^{2} \)
13 \( 1 + 0.641iT - 13T^{2} \)
17 \( 1 + (4.43 + 4.43i)T + 17iT^{2} \)
19 \( 1 + (1.72 - 1.72i)T - 19iT^{2} \)
23 \( 1 - 7.45iT - 23T^{2} \)
31 \( 1 + (-2.62 + 2.62i)T - 31iT^{2} \)
37 \( 1 + (-0.861 + 0.861i)T - 37iT^{2} \)
41 \( 1 + (-6.04 + 6.04i)T - 41iT^{2} \)
43 \( 1 + (-0.168 + 0.168i)T - 43iT^{2} \)
47 \( 1 + (-2.85 - 2.85i)T + 47iT^{2} \)
53 \( 1 - 8.27T + 53T^{2} \)
59 \( 1 + 4.23iT - 59T^{2} \)
61 \( 1 + (3.90 + 3.90i)T + 61iT^{2} \)
67 \( 1 + 5.32T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + (0.511 - 0.511i)T - 73iT^{2} \)
79 \( 1 + (2.35 - 2.35i)T - 79iT^{2} \)
83 \( 1 - 9.82iT - 83T^{2} \)
89 \( 1 + (-10.4 - 10.4i)T + 89iT^{2} \)
97 \( 1 + (-3.25 + 3.25i)T - 97iT^{2} \)
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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

−9.281323365337940368005135782589, −9.034200016767171799222185872315, −8.047800353127460501143633989514, −7.30549587155868154734544592178, −6.76603859245912358397278918190, −5.62061438894190697225472526454, −4.15901631245194229207308476278, −3.15074600856705741993056503697, −2.16333223390604129000454264366, −0.964006944316247242601851412547, 2.17719025029263952562649434100, 2.94058296092952535073453499914, 4.26482337372595005882809064531, 4.50131518385796157069631589942, 5.98419654208279388543630559305, 7.03732982300592021750597988379, 8.230330033573882378977126358837, 8.796021698223944347970817311212, 9.292926391591198373845712846782, 10.44731368867919030081113894760

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