Properties

Label 1-968-968.307-r0-0-0
Degree $1$
Conductor $968$
Sign $0.819 + 0.572i$
Analytic cond. $4.49537$
Root an. cond. $4.49537$
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  + 3-s + (0.142 + 0.989i)5-s + (−0.654 − 0.755i)7-s + 9-s + (0.415 + 0.909i)13-s + (0.142 + 0.989i)15-s + (0.959 − 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.654 − 0.755i)21-s + (0.654 − 0.755i)23-s + (−0.959 + 0.281i)25-s + 27-s + (−0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.654 − 0.755i)35-s + ⋯
L(s)  = 1  + 3-s + (0.142 + 0.989i)5-s + (−0.654 − 0.755i)7-s + 9-s + (0.415 + 0.909i)13-s + (0.142 + 0.989i)15-s + (0.959 − 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.654 − 0.755i)21-s + (0.654 − 0.755i)23-s + (−0.959 + 0.281i)25-s + 27-s + (−0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.654 − 0.755i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(4.49537\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (0:\ ),\ 0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.132475436 + 0.6715215896i\)
\(L(\frac12)\) \(\approx\) \(2.132475436 + 0.6715215896i\)
\(L(1)\) \(\approx\) \(1.517672379 + 0.2297989994i\)
\(L(1)\) \(\approx\) \(1.517672379 + 0.2297989994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.38635531101812823169864552994, −20.90118650795161634681904764775, −20.02977805002401669178196136712, −19.53226819687134125364810823121, −18.60348070659146395980830169001, −17.930464530343966851682076550705, −16.73482857546259415032161922272, −16.048499904942188810639040824801, −15.34064898146765697399050936608, −14.63727180714558420243727712338, −13.44623538600936548291709442021, −13.02888925926347715811373963990, −12.36007689526827391273718909539, −11.312153989984007560116860076185, −9.846555926780061080648107732504, −9.56234272088921054502647961422, −8.63102667341646156386451982868, −8.004100294802246933724640983082, −7.06408359337632781840187044539, −5.68936097358842348648898587204, −5.2089172484235736878634379267, −3.7637716908423543952677931251, −3.185478622560604459194271285321, −2.04117283523358718345371896162, −0.99282598022542139902117321407, 1.26241759365092300339987413920, 2.38638225788356613097982176124, 3.45768584695384274110244606755, 3.71400384842310806508605816910, 5.14703158908039862892474965099, 6.52937944246553070412366786779, 7.04936959190514648760919492717, 7.78763218294173421301975081732, 8.90952691309035999494165842189, 9.76670713388651200354040591171, 10.28673647265772520059702394336, 11.25137322360705228372246626902, 12.330389589753548839850458424123, 13.34868832105232617898020087450, 14.00721265306702678179305224371, 14.42875829342962443501097059970, 15.4089746174128197011388254531, 16.218826334077899674453357486417, 16.9760095837937978076845296470, 18.238243875423664072528434154485, 18.85259695440638228346451599101, 19.27130821543574961319098515501, 20.39030635542938502697524748501, 20.81535029297867756406126577388, 21.82615872017861523923531549294

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