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Properties

Label 2-1072-268.267-c1-0-33
Degree $2$
Conductor $1072$
Sign $-0.892 - 0.451i$
Analytic cond. $8.55996$
Root an. cond. $2.92574$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·3-s − 3i·5-s − 4.30·7-s − 1.30·9-s − 3·11-s + 6.90i·13-s − 3.90i·15-s + 4.60·17-s − 5.30i·19-s − 5.60·21-s + 7.60i·23-s − 4·25-s − 5.60·27-s − 7.60·29-s − 5.60·31-s + ⋯
L(s)  = 1  + 0.752·3-s − 1.34i·5-s − 1.62·7-s − 0.434·9-s − 0.904·11-s + 1.91i·13-s − 1.00i·15-s + 1.11·17-s − 1.21i·19-s − 1.22·21-s + 1.58i·23-s − 0.800·25-s − 1.07·27-s − 1.41·29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1072\)    =    \(2^{4} \cdot 67\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(8.55996\)
Root analytic conductor: \(2.92574\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1072} (1071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1072,\ (\ :1/2),\ -0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
67 \( 1 + (7.30 + 3.69i)T \)
good3 \( 1 - 1.30T + 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 4.30T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 6.90iT - 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 + 5.30iT - 19T^{2} \)
23 \( 1 - 7.60iT - 23T^{2} \)
29 \( 1 + 7.60T + 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + 3.90iT - 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 - 3.69iT - 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 9.21iT - 59T^{2} \)
61 \( 1 + 6.90iT - 61T^{2} \)
71 \( 1 + 1.60iT - 71T^{2} \)
73 \( 1 + 8.60T + 73T^{2} \)
79 \( 1 + 4.69T + 79T^{2} \)
83 \( 1 - 6.69iT - 83T^{2} \)
89 \( 1 - 4.39T + 89T^{2} \)
97 \( 1 + 4.81iT - 97T^{2} \)
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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

−9.303359792336877815329909050513, −8.941385137986559332984755013722, −7.81256344879696215039092656773, −7.05067955197044020744411027665, −5.87755188090651168987046566075, −5.12618434630317131118295190853, −3.89515604040611391609214914499, −3.10511369233147523544263599322, −1.83211212787709741401321177673, 0, 2.55737154256160772752495757173, 3.13497463474871162191898552945, 3.58669082613473774830850403064, 5.65341570126185990897593158003, 5.98901646140210541127984624392, 7.21933507847259645993663016496, 7.78873211455027837174816176862, 8.625216611769191668332353630770, 9.815710572284591103375859430554

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