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Properties

Label 1-2e7-128.21-r0-0-0
Degree $1$
Conductor $128$
Sign $-0.427 + 0.903i$
Analytic cond. $0.594429$
Root an. cond. $0.594429$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.195 + 0.980i)3-s + (−0.831 + 0.555i)5-s + (0.923 + 0.382i)7-s + (−0.923 + 0.382i)9-s + (−0.980 − 0.195i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.555 + 0.831i)19-s + (−0.195 + 0.980i)21-s + (−0.382 − 0.923i)23-s + (0.382 − 0.923i)25-s + (−0.555 − 0.831i)27-s + (0.980 − 0.195i)29-s + i·31-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)3-s + (−0.831 + 0.555i)5-s + (0.923 + 0.382i)7-s + (−0.923 + 0.382i)9-s + (−0.980 − 0.195i)11-s + (0.831 + 0.555i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (−0.555 + 0.831i)19-s + (−0.195 + 0.980i)21-s + (−0.382 − 0.923i)23-s + (0.382 − 0.923i)25-s + (−0.555 − 0.831i)27-s + (0.980 − 0.195i)29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.427 + 0.903i$
Analytic conductor: \(0.594429\)
Root analytic conductor: \(0.594429\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ -0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4984937027 + 0.7872075807i\)
\(L(\frac12)\) \(\approx\) \(0.4984937027 + 0.7872075807i\)
\(L(1)\) \(\approx\) \(0.8214208856 + 0.5058587313i\)
\(L(1)\) \(\approx\) \(0.8214208856 + 0.5058587313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.195 + 0.980i)T \)
5 \( 1 + (-0.831 + 0.555i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.980 - 0.195i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.555 + 0.831i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.980 - 0.195i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.555 + 0.831i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.195 - 0.980i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.980 + 0.195i)T \)
59 \( 1 + (0.831 - 0.555i)T \)
61 \( 1 + (-0.195 - 0.980i)T \)
67 \( 1 + (-0.195 - 0.980i)T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.555 - 0.831i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.43386308614503060738777228806, −27.60677044492538636209349268437, −26.42850149576630865273879936524, −25.350801695572593758067659039470, −24.20814654340805877768808845857, −23.69007971182269953805629771025, −22.873610887581870429968132455850, −21.04416522282136925015368351955, −20.26974692573787194453198443825, −19.42228650141525496329642354977, −18.13004435318834462376271497042, −17.54903533673274629279118413143, −16.02278685125344550478021260312, −15.05339301543639295158274267381, −13.64236694026771512706862348459, −12.935827919274105034466138380298, −11.68205333586736434284083111776, −10.8801137120692120010649474881, −8.87403964868979618116437230042, −7.972949536316028989334830689487, −7.23246511628088344838671570291, −5.54378512325489057586321178369, −4.22727330726436406002226196895, −2.517700731108094044528611480785, −0.872029738404476876207098219537, 2.402460388377880813060667872741, 3.822298921584754456663497580768, 4.79293732477461635116765064395, 6.27562689905740953801470068389, 8.112425950143340439749637309556, 8.58850390345041039682235542750, 10.43085421746936486621560470224, 10.97476328917083633143226182613, 12.11539337181789992561844777473, 13.8583201208245493477781070848, 14.8627007346825789648650751012, 15.58826516350604387244218871383, 16.51888530066229140699571921730, 18.00125594604347162957079264421, 18.928867906670481254053804139, 20.168289458353374321910139699531, 21.13004143694117669411744346761, 21.86090455178163892225531912940, 23.13518625415299239508304170628, 23.87103449199073817301157914682, 25.33458011006518045192337446109, 26.44103212267501311717159400908, 26.97422448053001753588750270326, 28.00426548472413420152433996285, 28.7649249853674064316151643144

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