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Properties

Label 1-667-667.110-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.165 - 0.986i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.339 − 0.940i)2-s + (−0.301 + 0.953i)3-s + (−0.768 − 0.639i)4-s + (−0.0611 + 0.998i)5-s + (0.794 + 0.607i)6-s + (0.917 − 0.396i)7-s + (−0.862 + 0.505i)8-s + (−0.818 − 0.574i)9-s + (0.917 + 0.396i)10-s + (0.101 − 0.994i)11-s + (0.841 − 0.540i)12-s + (−0.999 + 0.0407i)13-s + (−0.0611 − 0.998i)14-s + (−0.933 − 0.359i)15-s + (0.182 + 0.983i)16-s + (0.415 − 0.909i)17-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (−0.301 + 0.953i)3-s + (−0.768 − 0.639i)4-s + (−0.0611 + 0.998i)5-s + (0.794 + 0.607i)6-s + (0.917 − 0.396i)7-s + (−0.862 + 0.505i)8-s + (−0.818 − 0.574i)9-s + (0.917 + 0.396i)10-s + (0.101 − 0.994i)11-s + (0.841 − 0.540i)12-s + (−0.999 + 0.0407i)13-s + (−0.0611 − 0.998i)14-s + (−0.933 − 0.359i)15-s + (0.182 + 0.983i)16-s + (0.415 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.165 - 0.986i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.165 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6882301854 - 0.8133627462i\)
\(L(\frac12)\) \(\approx\) \(0.6882301854 - 0.8133627462i\)
\(L(1)\) \(\approx\) \(0.9235409295 - 0.3218234875i\)
\(L(1)\) \(\approx\) \(0.9235409295 - 0.3218234875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.339 - 0.940i)T \)
3 \( 1 + (-0.301 + 0.953i)T \)
5 \( 1 + (-0.0611 + 0.998i)T \)
7 \( 1 + (0.917 - 0.396i)T \)
11 \( 1 + (0.101 - 0.994i)T \)
13 \( 1 + (-0.999 + 0.0407i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (-0.768 - 0.639i)T \)
31 \( 1 + (0.996 - 0.0815i)T \)
37 \( 1 + (-0.818 - 0.574i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.996 + 0.0815i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.882 + 0.470i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (0.557 - 0.830i)T \)
67 \( 1 + (0.101 + 0.994i)T \)
71 \( 1 + (0.947 - 0.320i)T \)
73 \( 1 + (-0.979 + 0.202i)T \)
79 \( 1 + (0.182 - 0.983i)T \)
83 \( 1 + (0.742 - 0.670i)T \)
89 \( 1 + (-0.933 + 0.359i)T \)
97 \( 1 + (0.986 - 0.162i)T \)
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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

−23.25925962574884538787296632702, −22.50574792362182729702678710261, −21.45734018133793937443113937853, −20.72590976465112046194045148756, −19.6069575352497264412199870233, −18.74820051796305978262801709002, −17.697733257208958809596725372494, −17.22993902013260663763264771557, −16.747757258880889070356872962814, −15.42956955627502713079651429299, −14.74567919052011182548873801873, −13.92966207968062421462201664710, −12.90230871718800573172814500171, −12.24819228831660385335761735530, −11.9192405854056195709377045458, −10.25552266776796525219758827573, −9.00934713945838822358345284496, −8.169944490408056520939701815299, −7.70423053056256873238752589662, −6.656837502585849372630171815130, −5.6370092750379514626019684346, −4.96821429330656816661630035949, −4.16103204608121066795004657901, −2.361963202596166417350585337963, −1.30579778077615070821723850123, 0.514434477948534778318379653736, 2.28372524806758532881893012996, 3.12813880903792797162996774638, 4.05760401069807239350986072578, 4.91515047111898937856381076258, 5.73622526467603832156348836432, 6.95182266890336904476669031000, 8.29211994252508071642711534682, 9.27538368437449759113523469467, 10.25831327006430148784931571349, 10.759112818947265825245204657259, 11.506534712087543894735711417118, 12.06343280941398233797367829068, 13.67187529997661656291143489649, 14.22589489996507312570229876182, 14.875181308592910437905482994730, 15.709320513555176309776755794343, 17.071845165716361072100122894450, 17.601402593308073551644452074199, 18.640940599324372765049472549992, 19.38682676862278408371262208215, 20.29338420766695085501886065731, 21.148859734659202291637036246229, 21.69936223377152788028630237651, 22.29984458030698503459693052222

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