Properties

Label 2-2e7-128.3-c2-0-12
Degree $2$
Conductor $128$
Sign $0.995 + 0.0963i$
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  + (−1.84 − 0.777i)2-s + (1.79 − 1.47i)3-s + (2.78 + 2.86i)4-s + (5.80 + 1.76i)5-s + (−4.46 + 1.32i)6-s + (0.867 + 4.35i)7-s + (−2.91 − 7.45i)8-s + (−0.698 + 3.51i)9-s + (−9.32 − 7.75i)10-s + (1.96 + 19.9i)11-s + (9.25 + 1.03i)12-s + (−2.26 − 7.46i)13-s + (1.79 − 8.70i)14-s + (13.0 − 5.40i)15-s + (−0.433 + 15.9i)16-s + (11.0 − 26.5i)17-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)2-s + (0.599 − 0.492i)3-s + (0.697 + 0.716i)4-s + (1.16 + 0.352i)5-s + (−0.744 + 0.220i)6-s + (0.123 + 0.622i)7-s + (−0.363 − 0.931i)8-s + (−0.0776 + 0.390i)9-s + (−0.932 − 0.775i)10-s + (0.178 + 1.81i)11-s + (0.771 + 0.0865i)12-s + (−0.174 − 0.574i)13-s + (0.128 − 0.621i)14-s + (0.869 − 0.360i)15-s + (−0.0270 + 0.999i)16-s + (0.647 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.995 + 0.0963i$
Analytic conductor: \(3.48774\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.995 + 0.0963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34247 - 0.0648295i\)
\(L(\frac12)\) \(\approx\) \(1.34247 - 0.0648295i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.84 + 0.777i)T \)
good3 \( 1 + (-1.79 + 1.47i)T + (1.75 - 8.82i)T^{2} \)
5 \( 1 + (-5.80 - 1.76i)T + (20.7 + 13.8i)T^{2} \)
7 \( 1 + (-0.867 - 4.35i)T + (-45.2 + 18.7i)T^{2} \)
11 \( 1 + (-1.96 - 19.9i)T + (-118. + 23.6i)T^{2} \)
13 \( 1 + (2.26 + 7.46i)T + (-140. + 93.8i)T^{2} \)
17 \( 1 + (-11.0 + 26.5i)T + (-204. - 204. i)T^{2} \)
19 \( 1 + (-8.06 + 15.0i)T + (-200. - 300. i)T^{2} \)
23 \( 1 + (0.595 - 0.397i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (1.40 - 14.2i)T + (-824. - 164. i)T^{2} \)
31 \( 1 + (24.5 + 24.5i)T + 961iT^{2} \)
37 \( 1 + (31.9 + 59.8i)T + (-760. + 1.13e3i)T^{2} \)
41 \( 1 + (-9.33 + 6.23i)T + (643. - 1.55e3i)T^{2} \)
43 \( 1 + (18.2 + 14.9i)T + (360. + 1.81e3i)T^{2} \)
47 \( 1 + (33.2 - 80.2i)T + (-1.56e3 - 1.56e3i)T^{2} \)
53 \( 1 + (0.361 + 3.67i)T + (-2.75e3 + 548. i)T^{2} \)
59 \( 1 + (-2.11 - 0.641i)T + (2.89e3 + 1.93e3i)T^{2} \)
61 \( 1 + (-11.0 + 9.06i)T + (725. - 3.64e3i)T^{2} \)
67 \( 1 + (56.9 + 69.4i)T + (-875. + 4.40e3i)T^{2} \)
71 \( 1 + (48.0 - 9.55i)T + (4.65e3 - 1.92e3i)T^{2} \)
73 \( 1 + (-96.8 - 19.2i)T + (4.92e3 + 2.03e3i)T^{2} \)
79 \( 1 + (-25.4 - 61.4i)T + (-4.41e3 + 4.41e3i)T^{2} \)
83 \( 1 + (-13.2 - 7.08i)T + (3.82e3 + 5.72e3i)T^{2} \)
89 \( 1 + (-35.3 + 52.8i)T + (-3.03e3 - 7.31e3i)T^{2} \)
97 \( 1 + (52.0 - 52.0i)T - 9.40e3iT^{2} \)
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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

−12.93047373987220391792898854762, −12.12204902243952242640562229434, −10.80282281006698752036166013896, −9.656649381605732269995120586625, −9.204681730271572068676183014241, −7.66502849177042706870390154223, −7.01164703298814833956373267857, −5.29395935233710346420544533504, −2.71988905679554670598080235673, −1.92897346794071540480213684828, 1.41744097477151907691547252510, 3.47982795217970985152223930264, 5.60582858374822561912049890505, 6.46329465123371458348650864940, 8.192516618346288077006632632527, 8.858035015604141392377974877094, 9.871994624075519056962753641994, 10.54171693496085527002411279413, 11.86583495601140735558003625243, 13.56144439622757180629678712845

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