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Properties

Label 2-968-11.7-c0-0-1
Degree $2$
Conductor $968$
Sign $0.0560 - 0.998i$
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  + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)5-s + (1.34 − 0.437i)7-s + (−0.309 − 0.951i)15-s + (0.831 + 1.14i)17-s + 1.41i·21-s − 23-s + (−0.809 + 0.587i)27-s + (−0.809 − 0.587i)31-s + (−0.831 + 1.14i)35-s + (0.309 + 0.951i)37-s − 1.41i·43-s + (0.809 − 0.587i)49-s + (−1.34 + 0.437i)51-s + (0.309 + 0.951i)59-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)5-s + (1.34 − 0.437i)7-s + (−0.309 − 0.951i)15-s + (0.831 + 1.14i)17-s + 1.41i·21-s − 23-s + (−0.809 + 0.587i)27-s + (−0.809 − 0.587i)31-s + (−0.831 + 1.14i)35-s + (0.309 + 0.951i)37-s − 1.41i·43-s + (0.809 − 0.587i)49-s + (−1.34 + 0.437i)51-s + (0.309 + 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.0560 - 0.998i$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ 0.0560 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9194672337\)
\(L(\frac12)\) \(\approx\) \(0.9194672337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.44781039295252739501896378935, −9.935796356815212136996426776718, −8.653850844395604142085851973840, −7.82294678613158328617213124504, −7.33957599405351662728484509097, −5.95616984084919423972195366782, −5.02798958301002478672233356594, −4.13077820116991630063198439689, −3.58418886820362785195632855123, −1.79232013696017646061738422107, 1.04052006312900831254152696419, 2.21110414080452641908912603621, 3.84433360353964124388669971458, 4.86275841968163416794270232086, 5.61302499055982326930347013028, 6.76363277597471891071497302005, 7.82702010236782684703440734912, 7.925101541511993234645651953449, 9.009401566965570370341194512573, 9.976122158878763470259100163074

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