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Properties

Label 2-968-88.59-c0-0-1
Degree $2$
Conductor $968$
Sign $0.530 + 0.847i$
Analytic cond. $0.483094$
Root an. cond. $0.695050$
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  + (−0.809 − 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)6-s + (0.309 − 0.951i)8-s + (0.5 + 0.363i)9-s + 0.618·12-s + (−0.809 + 0.587i)16-s + (1.30 − 0.951i)17-s + (−0.190 − 0.587i)18-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.363i)24-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)6-s + (0.309 − 0.951i)8-s + (0.5 + 0.363i)9-s + 0.618·12-s + (−0.809 + 0.587i)16-s + (1.30 − 0.951i)17-s + (−0.190 − 0.587i)18-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.363i)24-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7960196297\)
\(L(\frac12)\) \(\approx\) \(0.7960196297\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good3 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.11259501613401412181343544611, −9.373536798114136276684583967655, −8.262963694085976211439185580020, −7.79467691462959932552402181505, −7.03717858337785769635968929019, −6.01394163467952430332041593955, −4.61147687887228281503387112531, −3.47053398743761201590989506766, −2.35681230772135851514002206607, −1.26256999222350926764932468331, 1.37718933754546050902248480483, 2.99591221449679033372099230474, 4.28080413336667920044770606805, 5.23692254945087158657633877156, 6.23188142340669280174983303432, 7.09149062697562172761097032021, 7.88021524417630060264403830909, 8.886083011078873444029899959842, 9.360197644572703001981767176846, 10.24047360438505063388513290703

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