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Properties

Label 2-2e7-8.5-c1-0-0
Degree $2$
Conductor $128$
Sign $-i$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·3-s − 5.00·9-s + 2.82i·11-s + 6·17-s − 8.48i·19-s + 5·25-s − 5.65i·27-s − 8.00·33-s − 6·41-s − 8.48i·43-s − 7·49-s + 16.9i·51-s + 24·57-s + 14.1i·59-s − 8.48i·67-s + ⋯
L(s)  = 1  + 1.63i·3-s − 1.66·9-s + 0.852i·11-s + 1.45·17-s − 1.94i·19-s + 25-s − 1.08i·27-s − 1.39·33-s − 0.937·41-s − 1.29i·43-s − 49-s + 2.37i·51-s + 3.17·57-s + 1.84i·59-s − 1.03i·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748881 + 0.748881i\)
\(L(\frac12)\) \(\approx\) \(0.748881 + 0.748881i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 8.48iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 8.48iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−13.82686932364188423095463472958, −12.44943601301639991206556802577, −11.29975411628232970704714828288, −10.35406548918393499584234425732, −9.588916288933705161532427497691, −8.664242019433138992834480113785, −7.08971988569328979813016041294, −5.35336554189979674456025177029, −4.48789758656622716181904592878, −3.08634036933586957896715177110, 1.38179389167272920808910458982, 3.23595485619640913873366119897, 5.58550133389113302414045527469, 6.51950332514357795625674257850, 7.76869240934951829307603994212, 8.380648284949726943323366842492, 10.00565361958677993380557336950, 11.36515988012176547866224364384, 12.29818754311778415065996876260, 12.92089563107896110519093354136

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