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Properties

Label 2-2e7-16.5-c11-0-26
Degree $2$
Conductor $128$
Sign $0.846 + 0.533i$
Analytic cond. $98.3479$
Root an. cond. $9.91705$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−255. + 255. i)3-s + (4.21e3 + 4.21e3i)5-s + 8.03e4i·7-s + 4.61e4i·9-s + (−6.11e5 − 6.11e5i)11-s + (−3.70e5 + 3.70e5i)13-s − 2.15e6·15-s − 7.25e5·17-s + (1.13e7 − 1.13e7i)19-s + (−2.05e7 − 2.05e7i)21-s − 4.89e7i·23-s − 1.33e7i·25-s + (−5.71e7 − 5.71e7i)27-s + (−4.98e7 + 4.98e7i)29-s + 4.20e7·31-s + ⋯
L(s)  = 1  + (−0.608 + 0.608i)3-s + (0.603 + 0.603i)5-s + 1.80i·7-s + 0.260i·9-s + (−1.14 − 1.14i)11-s + (−0.276 + 0.276i)13-s − 0.733·15-s − 0.123·17-s + (1.04 − 1.04i)19-s + (−1.09 − 1.09i)21-s − 1.58i·23-s − 0.272i·25-s + (−0.766 − 0.766i)27-s + (−0.451 + 0.451i)29-s + 0.263·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.846 + 0.533i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ 0.846 + 0.533i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.7741510360\)
\(L(\frac12)\) \(\approx\) \(0.7741510360\)
\(L(\frac{13}{2})\) 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 \( 1 + (255. - 255. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-4.21e3 - 4.21e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 8.03e4iT - 1.97e9T^{2} \)
11 \( 1 + (6.11e5 + 6.11e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (3.70e5 - 3.70e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 7.25e5T + 3.42e13T^{2} \)
19 \( 1 + (-1.13e7 + 1.13e7i)T - 1.16e14iT^{2} \)
23 \( 1 + 4.89e7iT - 9.52e14T^{2} \)
29 \( 1 + (4.98e7 - 4.98e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 4.20e7T + 2.54e16T^{2} \)
37 \( 1 + (-1.05e8 - 1.05e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 4.49e8iT - 5.50e17T^{2} \)
43 \( 1 + (7.89e8 + 7.89e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.32e9T + 2.47e18T^{2} \)
53 \( 1 + (1.74e9 + 1.74e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-6.71e8 - 6.71e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (-8.30e8 + 8.30e8i)T - 4.35e19iT^{2} \)
67 \( 1 + (3.88e9 - 3.88e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 5.14e9iT - 2.31e20T^{2} \)
73 \( 1 + 4.42e9iT - 3.13e20T^{2} \)
79 \( 1 - 2.71e10T + 7.47e20T^{2} \)
83 \( 1 + (1.81e10 - 1.81e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 5.56e10iT - 2.77e21T^{2} \)
97 \( 1 - 2.61e10T + 7.15e21T^{2} \)
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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

−11.06386537872754441161469751944, −10.23966742735446343995783368348, −9.139626487155950851340094482285, −8.156088526520229213565015828210, −6.47011463778363771619952740022, −5.54713635197089045376205628135, −4.93793512710053446810872691598, −2.88390978961896877316025214334, −2.30976575282560600544221680447, −0.21170977458062768485389966611, 0.923333905751545226710571592994, 1.69583410269594999834538718126, 3.57119524187516293425445129090, 4.88790751481026730609751782949, 5.85075416376817167427490257448, 7.31790651337327268574241573607, 7.61304818746486995874578813341, 9.597628555495421928175876435331, 10.15008183961988610609912943828, 11.34930219285390912134690180354

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