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Properties

Label 2-2e7-16.3-c2-0-0
Degree $2$
Conductor $128$
Sign $-0.835 + 0.548i$
Analytic cond. $3.48774$
Root an. cond. $1.86755$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.24 + 3.24i)3-s + (0.0586 − 0.0586i)5-s − 4.61·7-s − 12.1i·9-s + (−5.36 − 5.36i)11-s + (−11.0 − 11.0i)13-s + 0.381i·15-s − 12.8·17-s + (2.63 − 2.63i)19-s + (14.9 − 14.9i)21-s − 16.3·23-s + 24.9i·25-s + (10.1 + 10.1i)27-s + (26.0 + 26.0i)29-s + 20.2i·31-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)3-s + (0.0117 − 0.0117i)5-s − 0.659·7-s − 1.34i·9-s + (−0.487 − 0.487i)11-s + (−0.850 − 0.850i)13-s + 0.0254i·15-s − 0.757·17-s + (0.138 − 0.138i)19-s + (0.714 − 0.714i)21-s − 0.712·23-s + 0.999i·25-s + (0.374 + 0.374i)27-s + (0.898 + 0.898i)29-s + 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.835 + 0.548i$
Analytic conductor: \(3.48774\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.835 + 0.548i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0223296 - 0.0746983i\)
\(L(\frac12)\) \(\approx\) \(0.0223296 - 0.0746983i\)
\(L(2)\) 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 \( 1 + (3.24 - 3.24i)T - 9iT^{2} \)
5 \( 1 + (-0.0586 + 0.0586i)T - 25iT^{2} \)
7 \( 1 + 4.61T + 49T^{2} \)
11 \( 1 + (5.36 + 5.36i)T + 121iT^{2} \)
13 \( 1 + (11.0 + 11.0i)T + 169iT^{2} \)
17 \( 1 + 12.8T + 289T^{2} \)
19 \( 1 + (-2.63 + 2.63i)T - 361iT^{2} \)
23 \( 1 + 16.3T + 529T^{2} \)
29 \( 1 + (-26.0 - 26.0i)T + 841iT^{2} \)
31 \( 1 - 20.2iT - 961T^{2} \)
37 \( 1 + (41.2 - 41.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (0.786 + 0.786i)T + 1.84e3iT^{2} \)
47 \( 1 + 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (1.06 - 1.06i)T - 2.80e3iT^{2} \)
59 \( 1 + (-32.5 - 32.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (15.2 + 15.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (60.0 - 60.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 56.3T + 5.04e3T^{2} \)
73 \( 1 - 9.70iT - 5.32e3T^{2} \)
79 \( 1 - 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (-26.7 + 26.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−13.59457100564313397796853661676, −12.50469128883798171335742020407, −11.52400096566172668417186090139, −10.45846204892996218525419556716, −9.959712304663497769527905915647, −8.637158610548380785767479701277, −6.96099038159942127763572686515, −5.67726747359428109381844174437, −4.81759540745390455530580231329, −3.25433756816062707082103147225, 0.05694125492051162914524604093, 2.19939280454906160158206928061, 4.57970248803426317715879819060, 6.03255275298679639369569014070, 6.81864492273731737644749820026, 7.82750678637676371220000600477, 9.472944608661966246316782975857, 10.58066789309344965148544814633, 11.80002710435927985861563636580, 12.36497372818624964862458017936

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