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Properties

Label 2-2e7-16.13-c11-0-19
Degree $2$
Conductor $128$
Sign $0.194 + 0.980i$
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  + (−367. − 367. i)3-s + (502. − 502. i)5-s + 3.30e4i·7-s + 9.34e4i·9-s + (−3.35e5 + 3.35e5i)11-s + (−5.68e5 − 5.68e5i)13-s − 3.69e5·15-s − 6.78e6·17-s + (−1.33e6 − 1.33e6i)19-s + (1.21e7 − 1.21e7i)21-s − 1.07e6i·23-s + 4.83e7i·25-s + (−3.07e7 + 3.07e7i)27-s + (−5.01e7 − 5.01e7i)29-s + 2.95e8·31-s + ⋯
L(s)  = 1  + (−0.873 − 0.873i)3-s + (0.0718 − 0.0718i)5-s + 0.743i·7-s + 0.527i·9-s + (−0.628 + 0.628i)11-s + (−0.424 − 0.424i)13-s − 0.125·15-s − 1.15·17-s + (−0.124 − 0.124i)19-s + (0.649 − 0.649i)21-s − 0.0347i·23-s + 0.989i·25-s + (−0.412 + 0.412i)27-s + (−0.453 − 0.453i)29-s + 1.85·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.194 + 0.980i$
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: \((2,\ 128,\ (\ :11/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.7908530379\)
\(L(\frac12)\) \(\approx\) \(0.7908530379\)
\(L(\frac{13}{2})\) 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 \( 1 + (367. + 367. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-502. + 502. i)T - 4.88e7iT^{2} \)
7 \( 1 - 3.30e4iT - 1.97e9T^{2} \)
11 \( 1 + (3.35e5 - 3.35e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (5.68e5 + 5.68e5i)T + 1.79e12iT^{2} \)
17 \( 1 + 6.78e6T + 3.42e13T^{2} \)
19 \( 1 + (1.33e6 + 1.33e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 1.07e6iT - 9.52e14T^{2} \)
29 \( 1 + (5.01e7 + 5.01e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 2.95e8T + 2.54e16T^{2} \)
37 \( 1 + (5.11e8 - 5.11e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 9.31e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.27e9 + 1.27e9i)T - 9.29e17iT^{2} \)
47 \( 1 + 1.80e7T + 2.47e18T^{2} \)
53 \( 1 + (2.29e9 - 2.29e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-4.81e8 + 4.81e8i)T - 3.01e19iT^{2} \)
61 \( 1 + (1.30e9 + 1.30e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-9.42e9 - 9.42e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.74e10iT - 2.31e20T^{2} \)
73 \( 1 + 8.91e9iT - 3.13e20T^{2} \)
79 \( 1 + 4.07e10T + 7.47e20T^{2} \)
83 \( 1 + (1.16e10 + 1.16e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 7.96e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.03e10T + 7.15e21T^{2} \)
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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

−11.24901680897402213566512742244, −10.06729876535890033731377704501, −8.839766126446933108752031528713, −7.60378422312846902437950130757, −6.63214342095891018472980158998, −5.66480004997671416717865753815, −4.67153985885786647172818757243, −2.74420726791982830102789491627, −1.66307908079199235490956332195, −0.32490589055600600191407042044, 0.60231700485534954366221629313, 2.38750772314019593700117739129, 3.97973587332443431506915674939, 4.78496080549267674169900011359, 5.89832759044578718687939133317, 7.02348097499493595698257279499, 8.370733505474247856980964912476, 9.667272872127105502662919710178, 10.62385816086385267387255132177, 11.07708221324689056434257517779

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