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Properties

Label 2-1280-40.29-c1-0-33
Degree $2$
Conductor $1280$
Sign $0.601 + 0.798i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
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  + 2.90·3-s + (−2.21 − 0.311i)5-s − 3.52i·7-s + 5.42·9-s + 3.80i·11-s + 2.62·13-s + (−6.42 − 0.903i)15-s − 5.80i·17-s − 5.05i·19-s − 10.2i·21-s + 0.474i·23-s + (4.80 + 1.37i)25-s + 7.05·27-s − 2i·29-s + 2.75·31-s + ⋯
L(s)  = 1  + 1.67·3-s + (−0.990 − 0.139i)5-s − 1.33i·7-s + 1.80·9-s + 1.14i·11-s + 0.727·13-s + (−1.65 − 0.233i)15-s − 1.40i·17-s − 1.15i·19-s − 2.23i·21-s + 0.0989i·23-s + (0.961 + 0.275i)25-s + 1.35·27-s − 0.371i·29-s + 0.494·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.601 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.554958609\)
\(L(\frac12)\) \(\approx\) \(2.554958609\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.21 + 0.311i)T \)
good3 \( 1 - 2.90T + 3T^{2} \)
7 \( 1 + 3.52iT - 7T^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 + 5.80iT - 17T^{2} \)
19 \( 1 + 5.05iT - 19T^{2} \)
23 \( 1 - 0.474iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 + 5.18T + 41T^{2} \)
43 \( 1 - 1.95T + 43T^{2} \)
47 \( 1 + 5.33iT - 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 + 5.05iT - 59T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 - 7.76T + 67T^{2} \)
71 \( 1 + 4.85T + 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 + 5.24T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−9.458118704510396991516071173532, −8.670629947767395939186536790409, −7.87845505347878713270294924855, −7.27597744434109964299112139178, −6.85109637050153044217467026326, −4.77423615118698125310883709925, −4.22482062425475065954444429449, −3.41603409681711236840952923078, −2.47689149790938805736249372421, −0.958793902544414575486056163560, 1.62022215099277919971756635414, 2.86055837631765748405711725966, 3.46454143679001584635421554695, 4.23425116868559581170039884697, 5.72826570012851736309181682187, 6.51886367164325885172790554656, 7.957213847324558914571234612039, 8.175331327710960741105278410109, 8.712868539068696738250107173176, 9.448393209588550181501447089129

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