Properties

Label 2-968-11.8-c0-0-1
Degree $2$
Conductor $968$
Sign $-0.996 - 0.0780i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.996 - 0.0780i$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ -0.996 - 0.0780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3668998776\)
\(L(\frac12)\) \(\approx\) \(0.3668998776\)
\(L(1)\) 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

−9.806902537093754993139759222945, −8.845196552180715528560813581608, −8.019672360615319776833000438530, −7.14826837206461467212657975032, −6.52752606886101678762747676114, −5.72028311805651190275004671549, −4.21668534274142585456052731911, −3.61729960179301366850380031045, −1.95909544567123710920095841052, −0.34730094615671320267926251175, 2.59608908758388173208369220750, 3.56303110613607760298212309982, 4.32801509692024875467776243308, 5.40749319306149873412100868965, 6.42463895595803678770560707106, 7.17167474085462440103118300096, 8.142946912133111121694177450496, 9.513785034998994423379774977166, 9.560059818205856612827010617372, 10.65920248623109318207834675798

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