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Properties

Label 2-968-11.9-c1-0-10
Degree $2$
Conductor $968$
Sign $0.353 - 0.935i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−2.07 + 1.50i)3-s + (−0.173 − 0.534i)5-s + (4.14 + 3.01i)7-s + (1.10 − 3.38i)9-s + (0.965 − 2.97i)13-s + (1.16 + 0.845i)15-s + (0.618 + 1.90i)17-s + (3.23 − 2.35i)19-s − 13.1·21-s + 6.56·23-s + (3.78 − 2.75i)25-s + (0.444 + 1.36i)27-s + (−2.52 − 1.83i)29-s + (−0.444 + 1.36i)31-s + (0.889 − 2.73i)35-s + ⋯
L(s)  = 1  + (−1.19 + 0.869i)3-s + (−0.0776 − 0.238i)5-s + (1.56 + 1.13i)7-s + (0.366 − 1.12i)9-s + (0.267 − 0.823i)13-s + (0.300 + 0.218i)15-s + (0.149 + 0.461i)17-s + (0.742 − 0.539i)19-s − 2.86·21-s + 1.36·23-s + (0.757 − 0.550i)25-s + (0.0855 + 0.263i)27-s + (−0.469 − 0.340i)29-s + (−0.0798 + 0.245i)31-s + (0.150 − 0.462i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.353 - 0.935i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06026 + 0.732460i\)
\(L(\frac12)\) \(\approx\) \(1.06026 + 0.732460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (2.07 - 1.50i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.173 + 0.534i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-4.14 - 3.01i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.965 + 2.97i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.618 - 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 + (2.52 + 1.83i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.444 - 1.36i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.78 - 2.02i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.76 - 4.18i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.31 - 4.03i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.3 - 7.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.20 + 6.77i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.43T + 67T^{2} \)
71 \( 1 + (-1.13 - 3.50i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.52 - 1.83i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.889 - 2.73i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.81 - 8.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.68T + 89T^{2} \)
97 \( 1 + (-3.53 + 10.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37845428920253536833226558595, −9.397526902040438718004295574111, −8.549776889589737030765596556557, −7.85439019863193237947426909751, −6.50757814504500573613008712086, −5.37345592090534535944262357075, −5.21697032294634899252999336604, −4.33828955342745561643425585576, −2.83283192850721620042852544810, −1.17732683180085194569097248638, 0.916451685727645924495666785726, 1.75759396597092450944660111848, 3.65442169413885613236627693745, 4.89121203275185936249228405448, 5.37052757424540786740664496399, 6.72143273984476012167578451136, 7.16460702210297621907139939222, 7.84281572630730828985694198702, 8.920055588345960901840834219702, 10.16422148937239774457248708155

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