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Properties

Label 2-2e7-16.3-c10-0-14
Degree $2$
Conductor $128$
Sign $0.891 + 0.452i$
Analytic cond. $81.3257$
Root an. cond. $9.01807$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (46.0 − 46.0i)3-s + (−1.93e3 + 1.93e3i)5-s − 2.59e4·7-s + 5.48e4i·9-s + (1.23e4 + 1.23e4i)11-s + (−3.98e5 − 3.98e5i)13-s + 1.78e5i·15-s − 2.30e6·17-s + (3.56e5 − 3.56e5i)19-s + (−1.19e6 + 1.19e6i)21-s − 7.31e6·23-s + 2.28e6i·25-s + (5.24e6 + 5.24e6i)27-s + (1.42e7 + 1.42e7i)29-s − 3.35e7i·31-s + ⋯
L(s)  = 1  + (0.189 − 0.189i)3-s + (−0.618 + 0.618i)5-s − 1.54·7-s + 0.928i·9-s + (0.0766 + 0.0766i)11-s + (−1.07 − 1.07i)13-s + 0.234i·15-s − 1.62·17-s + (0.144 − 0.144i)19-s + (−0.292 + 0.292i)21-s − 1.13·23-s + 0.234i·25-s + (0.365 + 0.365i)27-s + (0.696 + 0.696i)29-s − 1.17i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(81.3257\)
Root analytic conductor: \(9.01807\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :5),\ 0.891 + 0.452i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.5603041233\)
\(L(\frac12)\) \(\approx\) \(0.5603041233\)
\(L(6)\) 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 \( 1 + (-46.0 + 46.0i)T - 5.90e4iT^{2} \)
5 \( 1 + (1.93e3 - 1.93e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 2.59e4T + 2.82e8T^{2} \)
11 \( 1 + (-1.23e4 - 1.23e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (3.98e5 + 3.98e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 2.30e6T + 2.01e12T^{2} \)
19 \( 1 + (-3.56e5 + 3.56e5i)T - 6.13e12iT^{2} \)
23 \( 1 + 7.31e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.42e7 - 1.42e7i)T + 4.20e14iT^{2} \)
31 \( 1 + 3.35e7iT - 8.19e14T^{2} \)
37 \( 1 + (-1.40e7 + 1.40e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.71e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.08e8 - 1.08e8i)T + 2.16e16iT^{2} \)
47 \( 1 - 2.53e8iT - 5.25e16T^{2} \)
53 \( 1 + (1.86e8 - 1.86e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-4.10e8 - 4.10e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (8.65e8 + 8.65e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (-1.18e9 + 1.18e9i)T - 1.82e18iT^{2} \)
71 \( 1 + 8.95e8T + 3.25e18T^{2} \)
73 \( 1 + 1.21e8iT - 4.29e18T^{2} \)
79 \( 1 - 3.49e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.04e8 + 1.04e8i)T - 1.55e19iT^{2} \)
89 \( 1 + 8.97e9iT - 3.11e19T^{2} \)
97 \( 1 - 4.31e9T + 7.37e19T^{2} \)
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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

−11.22021797737461259191394312344, −10.30646966801781823313278262849, −9.349591708540950340206398374309, −7.924901869475787100287055322658, −7.12740388269376595231191625608, −6.06558848171029778420893318464, −4.51856816710038922708814347329, −3.14986111311079019290272420165, −2.35912399083906954814467258844, −0.25417335816864063186296711250, 0.48265898469905548921848606170, 2.37050258384845036793352222876, 3.70914647950149748225861517554, 4.52262160802088427645843168910, 6.28269773794985362148974302904, 7.00025761575181402260072924263, 8.597941024343029444431230923535, 9.340956417257475470579646564395, 10.17585670216165147349685410506, 11.85866049745398674793883725706

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