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Properties

Label 2-2e8-32.29-c1-0-0
Degree $2$
Conductor $256$
Sign $-0.344 - 0.938i$
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  + (−1.27 − 0.529i)3-s + (0.707 + 1.70i)5-s + (−2.74 + 2.74i)7-s + (−0.766 − 0.766i)9-s + (−0.135 + 0.0560i)11-s + (−1.18 + 2.85i)13-s − 2.55i·15-s + 6.44i·17-s + (−0.805 + 1.94i)19-s + (4.97 − 2.05i)21-s + (0.749 + 0.749i)23-s + (1.12 − 1.12i)25-s + (2.16 + 5.22i)27-s + (−4.32 − 1.79i)29-s + 1.17·31-s + ⋯
L(s)  = 1  + (−0.738 − 0.305i)3-s + (0.316 + 0.763i)5-s + (−1.03 + 1.03i)7-s + (−0.255 − 0.255i)9-s + (−0.0408 + 0.0169i)11-s + (−0.327 + 0.790i)13-s − 0.660i·15-s + 1.56i·17-s + (−0.184 + 0.445i)19-s + (1.08 − 0.449i)21-s + (0.156 + 0.156i)23-s + (0.224 − 0.224i)25-s + (0.416 + 1.00i)27-s + (−0.802 − 0.332i)29-s + 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.344 - 0.938i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ -0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367648 + 0.526519i\)
\(L(\frac12)\) \(\approx\) \(0.367648 + 0.526519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 0.529i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.74 - 2.74i)T - 7iT^{2} \)
11 \( 1 + (0.135 - 0.0560i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.18 - 2.85i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.44iT - 17T^{2} \)
19 \( 1 + (0.805 - 1.94i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.749 - 0.749i)T + 23iT^{2} \)
29 \( 1 + (4.32 + 1.79i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (1.73 + 4.18i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.49 + 2.49i)T + 41iT^{2} \)
43 \( 1 + (-6.10 + 2.52i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.66iT - 47T^{2} \)
53 \( 1 + (1.64 - 0.682i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.43 + 3.47i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.46 - 1.43i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-14.0 - 5.83i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (3.40 - 3.40i)T - 71iT^{2} \)
73 \( 1 + (0.442 + 0.442i)T + 73iT^{2} \)
79 \( 1 - 7.07iT - 79T^{2} \)
83 \( 1 + (2.99 - 7.23i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.21 - 4.21i)T - 89iT^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−12.36872044920334394511779017554, −11.42835336025893855065105723149, −10.48571203566622630603038847391, −9.508144539048478172085729729630, −8.565438427617724915634003654124, −6.96762693944913221780927340845, −6.23497238796652464004458303946, −5.59925232228454296976831396087, −3.68139570922793666604560934220, −2.27898756332135788418641654694, 0.53096362411063449642581717770, 3.03230263626706055405403832585, 4.63844521708678572594800840059, 5.42002612019201886543616068482, 6.63328289818843335695116298220, 7.67297273483000441131285554985, 9.075116942435798586541697195343, 9.905623370724727107370722186942, 10.71569659607716858894917658169, 11.64771761233867640929883209271

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