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Properties

Label 1-2668-2668.1167-r0-0-0
Degree $1$
Conductor $2668$
Sign $-0.258 - 0.965i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (−0.101 − 0.994i)3-s + (−0.0203 − 0.999i)5-s + (−0.377 − 0.925i)7-s + (−0.979 + 0.202i)9-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.992 + 0.122i)15-s + (0.142 + 0.989i)17-s + (0.685 + 0.728i)19-s + (−0.882 + 0.470i)21-s + (−0.999 + 0.0407i)25-s + (0.301 + 0.953i)27-s + (0.523 − 0.852i)31-s + (0.377 − 0.925i)33-s + (−0.917 + 0.396i)35-s + ⋯
L(s)  = 1  + (−0.101 − 0.994i)3-s + (−0.0203 − 0.999i)5-s + (−0.377 − 0.925i)7-s + (−0.979 + 0.202i)9-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.992 + 0.122i)15-s + (0.142 + 0.989i)17-s + (0.685 + 0.728i)19-s + (−0.882 + 0.470i)21-s + (−0.999 + 0.0407i)25-s + (0.301 + 0.953i)27-s + (0.523 − 0.852i)31-s + (0.377 − 0.925i)33-s + (−0.917 + 0.396i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (1167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9763977041 - 1.272681609i\)
\(L(\frac12)\) \(\approx\) \(0.9763977041 - 1.272681609i\)
\(L(1)\) \(\approx\) \(0.9334452233 - 0.5292224946i\)
\(L(1)\) \(\approx\) \(0.9334452233 - 0.5292224946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.101 - 0.994i)T \)
5 \( 1 + (-0.0203 - 0.999i)T \)
7 \( 1 + (-0.377 - 0.925i)T \)
11 \( 1 + (0.882 + 0.470i)T \)
13 \( 1 + (0.488 + 0.872i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.685 + 0.728i)T \)
31 \( 1 + (0.523 - 0.852i)T \)
37 \( 1 + (0.979 - 0.202i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.523 - 0.852i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.986 + 0.162i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.947 - 0.320i)T \)
67 \( 1 + (0.882 - 0.470i)T \)
71 \( 1 + (0.591 - 0.806i)T \)
73 \( 1 + (0.557 - 0.830i)T \)
79 \( 1 + (-0.0611 - 0.998i)T \)
83 \( 1 + (0.970 + 0.242i)T \)
89 \( 1 + (0.992 + 0.122i)T \)
97 \( 1 + (0.452 + 0.891i)T \)
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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

−19.694820571755551042871580594772, −18.71321784248678885546713835742, −18.10014225733643388834062995026, −17.51694189192566643304908706106, −16.45718605126460183110231415489, −15.89769176966828423334788273086, −15.33821717936616083741917294191, −14.67883411209148106215426350648, −14.049504370498951764004147608383, −13.2482684203987943043719333905, −12.09916564487387364014435711505, −11.39961106666248419771302408701, −11.09060003758178866813541831362, −9.961449951817544671319830253160, −9.60819017398044324645649543285, −8.77157399659282610601238430894, −8.03605656850304023360330685875, −6.87208515544452329370250703143, −6.20826572279148046040817329606, −5.534816235624666475747414803153, −4.73689138859640428132405918528, −3.5932051710978291334983700844, −3.07172045828862480746652582804, −2.52308559903773509556226321413, −0.88405545445697769822888310991, 0.69605964931893069068419974589, 1.42940568622137917381654799347, 2.05717157342160012012525112821, 3.565930577698950903378203128801, 4.07954672121548457559792841891, 5.03361758842266294009228194657, 6.19345911765606276558740864672, 6.42243346246171299498135419556, 7.5926501430299627032379025967, 7.93794360249339355386633397429, 9.01210885082027368383767451707, 9.48970922595428254471293662681, 10.525734775495407624230955819220, 11.446719254943069761935823067758, 12.094431780453290958855879418, 12.66937379035480791137753983985, 13.37396976133173622876731477627, 13.97053373336277171113245105922, 14.5821374603899070135797125793, 15.763241356063662713397650403831, 16.58065953977198080554008139684, 17.028470614631884607986660456154, 17.47475088633403341062788187647, 18.48313151507862054209502177585, 19.24694400428454528130627408370

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