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Properties

Label 2-2e7-8.5-c3-0-1
Degree $2$
Conductor $128$
Sign $-i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
Motivic weight $3$
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 + 19·9-s + 70.7i·11-s − 90·17-s + 127. i·19-s + 125·25-s + 130. i·27-s − 200.·33-s + 522·41-s − 483. i·43-s − 343·49-s − 254. i·51-s − 360.·57-s − 325. i·59-s − 1.09e3i·67-s + ⋯
L(s)  = 1  + 0.544i·3-s + 0.703·9-s + 1.93i·11-s − 1.28·17-s + 1.53i·19-s + 25-s + 0.927i·27-s − 1.05·33-s + 1.98·41-s − 1.71i·43-s − 49-s − 0.698i·51-s − 0.836·57-s − 0.717i·59-s − 1.99i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06949 + 1.06949i\)
\(L(\frac12)\) \(\approx\) \(1.06949 + 1.06949i\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.82iT - 27T^{2} \)
5 \( 1 - 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 70.7iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 90T + 4.91e3T^{2} \)
19 \( 1 - 127. iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 522T + 6.89e4T^{2} \)
43 \( 1 + 483. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 325. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 1.09e3iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 430T + 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 681. iT - 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 - 1.91e3T + 9.12e5T^{2} \)
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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

−12.86613391569272739481506734153, −12.32145352280495509802959818033, −10.84174934320969070722998291614, −9.996913186961124265760855940422, −9.162901398394641071451612150416, −7.65244323634178350286896130015, −6.64124692147669390435640856039, −4.91964601291452429483575426540, −4.01134026870600113444691267406, −1.94800741825499369940914819677, 0.819017669647580252860004126787, 2.79117855394793387110811803056, 4.50670355090226620700501858155, 6.10170787573384112802259291180, 7.05336301730707782102125372761, 8.359883756521301568888731607607, 9.267765683929770566087148110692, 10.83996791860317420113309508985, 11.42036153002511934762883501937, 12.94212640267255725999332712455

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