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Properties

Label 2-2e7-16.13-c11-0-20
Degree $2$
Conductor $128$
Sign $0.373 - 0.927i$
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  + (296. + 296. i)3-s + (2.86e3 − 2.86e3i)5-s + 8.38e3i·7-s − 1.24e3i·9-s + (−8.45e4 + 8.45e4i)11-s + (3.85e5 + 3.85e5i)13-s + 1.70e6·15-s + 1.56e6·17-s + (−9.59e6 − 9.59e6i)19-s + (−2.48e6 + 2.48e6i)21-s + 3.88e7i·23-s + 3.23e7i·25-s + (5.29e7 − 5.29e7i)27-s + (8.81e7 + 8.81e7i)29-s − 7.23e7·31-s + ⋯
L(s)  = 1  + (0.704 + 0.704i)3-s + (0.410 − 0.410i)5-s + 0.188i·7-s − 0.00701i·9-s + (−0.158 + 0.158i)11-s + (0.287 + 0.287i)13-s + 0.578·15-s + 0.267·17-s + (−0.888 − 0.888i)19-s + (−0.132 + 0.132i)21-s + 1.25i·23-s + 0.663i·25-s + (0.709 − 0.709i)27-s + (0.797 + 0.797i)29-s − 0.453·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.373 - 0.927i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ 0.373 - 0.927i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.010617249\)
\(L(\frac12)\) \(\approx\) \(3.010617249\)
\(L(\frac{13}{2})\) 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 \( 1 + (-296. - 296. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-2.86e3 + 2.86e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 8.38e3iT - 1.97e9T^{2} \)
11 \( 1 + (8.45e4 - 8.45e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (-3.85e5 - 3.85e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.56e6T + 3.42e13T^{2} \)
19 \( 1 + (9.59e6 + 9.59e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 3.88e7iT - 9.52e14T^{2} \)
29 \( 1 + (-8.81e7 - 8.81e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 7.23e7T + 2.54e16T^{2} \)
37 \( 1 + (-9.98e7 + 9.98e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.13e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.05e9 + 1.05e9i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.33e9T + 2.47e18T^{2} \)
53 \( 1 + (2.50e9 - 2.50e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-3.81e9 + 3.81e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-7.73e9 - 7.73e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (2.91e9 + 2.91e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.77e10iT - 2.31e20T^{2} \)
73 \( 1 + 2.04e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.60e10T + 7.47e20T^{2} \)
83 \( 1 + (5.73e9 + 5.73e9i)T + 1.28e21iT^{2} \)
89 \( 1 - 1.07e10iT - 2.77e21T^{2} \)
97 \( 1 + 7.36e9T + 7.15e21T^{2} \)
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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

−11.30168190364795251524994546774, −10.15940703390323968179712184068, −9.197482649387084564702298308073, −8.680750292391198220364753795149, −7.24729266446908897554599408438, −5.87182796595421575247973932801, −4.68948752541758312244290828508, −3.59777105038931525063951326833, −2.43081207468753158888084747839, −1.07109580749422682155205266003, 0.64962001412466909962670898747, 1.99997389423518809415858681257, 2.78138437596813111119140872597, 4.20674697831728693084541683221, 5.81139281608342168073331990393, 6.81454838367732234369345132184, 7.950804092418412073287430815464, 8.647114101720683468367544498855, 10.11558836950883943324138101667, 10.82227222375570514313381671231

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