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Properties

Label 2-2e7-16.11-c2-0-2
Degree $2$
Conductor $128$
Sign $0.879 + 0.475i$
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  + (−2.10 − 2.10i)3-s + (4.62 + 4.62i)5-s + 3.04·7-s − 0.156i·9-s + (9.15 − 9.15i)11-s + (5.78 − 5.78i)13-s − 19.4i·15-s + 17.6·17-s + (1.15 + 1.15i)19-s + (−6.41 − 6.41i)21-s − 3.45·23-s + 17.8i·25-s + (−19.2 + 19.2i)27-s + (−12.1 + 12.1i)29-s + 38.5i·31-s + ⋯
L(s)  = 1  + (−0.700 − 0.700i)3-s + (0.925 + 0.925i)5-s + 0.435·7-s − 0.0174i·9-s + (0.831 − 0.831i)11-s + (0.444 − 0.444i)13-s − 1.29i·15-s + 1.03·17-s + (0.0606 + 0.0606i)19-s + (−0.305 − 0.305i)21-s − 0.150·23-s + 0.712i·25-s + (−0.713 + 0.713i)27-s + (−0.420 + 0.420i)29-s + 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(3.48774\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.879 + 0.475i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36032 - 0.343811i\)
\(L(\frac12)\) \(\approx\) \(1.36032 - 0.343811i\)
\(L(2)\) 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 \( 1 + (2.10 + 2.10i)T + 9iT^{2} \)
5 \( 1 + (-4.62 - 4.62i)T + 25iT^{2} \)
7 \( 1 - 3.04T + 49T^{2} \)
11 \( 1 + (-9.15 + 9.15i)T - 121iT^{2} \)
13 \( 1 + (-5.78 + 5.78i)T - 169iT^{2} \)
17 \( 1 - 17.6T + 289T^{2} \)
19 \( 1 + (-1.15 - 1.15i)T + 361iT^{2} \)
23 \( 1 + 3.45T + 529T^{2} \)
29 \( 1 + (12.1 - 12.1i)T - 841iT^{2} \)
31 \( 1 - 38.5iT - 961T^{2} \)
37 \( 1 + (-0.0972 - 0.0972i)T + 1.36e3iT^{2} \)
41 \( 1 + 51.5iT - 1.68e3T^{2} \)
43 \( 1 + (-1.70 + 1.70i)T - 1.84e3iT^{2} \)
47 \( 1 - 24.1iT - 2.20e3T^{2} \)
53 \( 1 + (27.0 + 27.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (19.5 - 19.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (16.7 - 16.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-75.8 - 75.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 134.T + 5.04e3T^{2} \)
73 \( 1 - 112. iT - 5.32e3T^{2} \)
79 \( 1 + 135. iT - 6.24e3T^{2} \)
83 \( 1 + (74.9 + 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 31.4iT - 7.92e3T^{2} \)
97 \( 1 - 31.5T + 9.40e3T^{2} \)
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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

−12.99089949100186247026584755652, −11.92271887836510699790458366540, −11.05950358171608345016098527175, −10.12011320420431323978726231439, −8.786728970198052002445738786765, −7.30277907420503591490248588390, −6.28367410475653378612314518168, −5.58300518907927942472752559811, −3.33220617384436717732479646181, −1.37200512943531283155049211805, 1.63092221800856725671928892897, 4.25310878954247218975445004319, 5.20086074039277086701014650559, 6.20380010533115955991526629571, 7.919722320660645743591798496796, 9.356206567854527360570921938045, 9.874992710115441382414829855254, 11.16340869597476135800282252352, 12.04605702857958146999359263855, 13.15631044524913406929623245655

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