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Properties

Label 2-2e8-16.5-c1-0-4
Degree $2$
Conductor $256$
Sign $0.130 + 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.130 + 0.991i$
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: \((2,\ 256,\ (\ :1/2),\ 0.130 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364399 - 0.319569i\)
\(L(\frac12)\) \(\approx\) \(0.364399 - 0.319569i\)
\(L(\frac{3}{2})\) 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

−11.52922859634112044372931733265, −10.88395393726862080951050268311, −10.20643049179719636240599031263, −8.918949602986920840160030253421, −8.086148303146771650391461229271, −6.54528370556173041753760761299, −5.39533187985337212972111640896, −4.55461713757027867490312322162, −3.59343735158927373750011230928, −0.43944973182846933218036722433, 1.81936575238758996839564970590, 3.71575959056194580032055335147, 5.33263924945624925992065660845, 6.42376764166804081698953806966, 7.08619027135601439682752384921, 7.992221127440982011782439258564, 9.340730389232682065941633272677, 10.93209636668021337667036913676, 11.34271098683219718872689370007, 11.99827800287203958346905697746

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