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Properties

Label 2-2e8-16.5-c1-0-5
Degree $2$
Conductor $256$
Sign $0.793 + 0.608i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
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  + (1.93 − 1.93i)3-s + (1.73 + 1.73i)5-s − 1.03i·7-s − 4.46i·9-s + (0.896 + 0.896i)11-s + (−3.73 + 3.73i)13-s + 6.69·15-s − 3.46·17-s + (−0.896 + 0.896i)19-s + (−1.99 − 1.99i)21-s − 6.69i·23-s + 0.999i·25-s + (−2.82 − 2.82i)27-s + (1.73 − 1.73i)29-s − 5.65·31-s + ⋯
L(s)  = 1  + (1.11 − 1.11i)3-s + (0.774 + 0.774i)5-s − 0.391i·7-s − 1.48i·9-s + (0.270 + 0.270i)11-s + (−1.03 + 1.03i)13-s + 1.72·15-s − 0.840·17-s + (−0.205 + 0.205i)19-s + (−0.436 − 0.436i)21-s − 1.39i·23-s + 0.199i·25-s + (−0.544 − 0.544i)27-s + (0.321 − 0.321i)29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75809 - 0.596793i\)
\(L(\frac12)\) \(\approx\) \(1.75809 - 0.596793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1.93 + 1.93i)T - 3iT^{2} \)
5 \( 1 + (-1.73 - 1.73i)T + 5iT^{2} \)
7 \( 1 + 1.03iT - 7T^{2} \)
11 \( 1 + (-0.896 - 0.896i)T + 11iT^{2} \)
13 \( 1 + (3.73 - 3.73i)T - 13iT^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + (0.896 - 0.896i)T - 19iT^{2} \)
23 \( 1 + 6.69iT - 23T^{2} \)
29 \( 1 + (-1.73 + 1.73i)T - 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-0.267 - 0.267i)T + 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (-5.79 - 5.79i)T + 43iT^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + (-4.26 - 4.26i)T + 53iT^{2} \)
59 \( 1 + (7.58 + 7.58i)T + 59iT^{2} \)
61 \( 1 + (-0.267 + 0.267i)T - 61iT^{2} \)
67 \( 1 + (2.96 - 2.96i)T - 67iT^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 - 9.46iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (5.79 - 5.79i)T - 83iT^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + 3.46T + 97T^{2} \)
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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

−12.19148418825548488192470325191, −10.93291999202350371988562216462, −9.811262094479631623076389723340, −9.006356631140951349996851290100, −7.87951052396593558132910707086, −6.88294011797325370910495329732, −6.42816171913845783294230460045, −4.40742518223259501489448033198, −2.73109019078727698079115608358, −1.92363720144388450694252278766, 2.22751409905012043109948860848, 3.51149908649555035306620279972, 4.84352508277310901171769690462, 5.65051792095341727937243508453, 7.45401611761923354667222081065, 8.687746076229629896044430575585, 9.178471912212720684099486029482, 9.915679864526637141091218359651, 10.86210927199404568926510662243, 12.26541005371930056177527321477

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