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Properties

Label 1-667-667.10-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.585 - 0.810i$
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  + (0.830 + 0.557i)2-s + (−0.242 + 0.970i)3-s + (0.377 + 0.925i)4-s + (0.262 + 0.965i)5-s + (−0.742 + 0.670i)6-s + (−0.947 + 0.320i)7-s + (−0.202 + 0.979i)8-s + (−0.882 − 0.470i)9-s + (−0.320 + 0.947i)10-s + (0.0815 − 0.996i)11-s + (−0.989 + 0.142i)12-s + (−0.339 − 0.940i)13-s + (−0.965 − 0.262i)14-s + (−0.999 + 0.0203i)15-s + (−0.714 + 0.699i)16-s + (0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (0.830 + 0.557i)2-s + (−0.242 + 0.970i)3-s + (0.377 + 0.925i)4-s + (0.262 + 0.965i)5-s + (−0.742 + 0.670i)6-s + (−0.947 + 0.320i)7-s + (−0.202 + 0.979i)8-s + (−0.882 − 0.470i)9-s + (−0.320 + 0.947i)10-s + (0.0815 − 0.996i)11-s + (−0.989 + 0.142i)12-s + (−0.339 − 0.940i)13-s + (−0.965 − 0.262i)14-s + (−0.999 + 0.0203i)15-s + (−0.714 + 0.699i)16-s + (0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5318122587 + 1.040348850i\)
\(L(\frac12)\) \(\approx\) \(-0.5318122587 + 1.040348850i\)
\(L(1)\) \(\approx\) \(0.6631135856 + 0.9776142489i\)
\(L(1)\) \(\approx\) \(0.6631135856 + 0.9776142489i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.830 + 0.557i)T \)
3 \( 1 + (-0.242 + 0.970i)T \)
5 \( 1 + (0.262 + 0.965i)T \)
7 \( 1 + (-0.947 + 0.320i)T \)
11 \( 1 + (0.0815 - 0.996i)T \)
13 \( 1 + (-0.339 - 0.940i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.925 + 0.377i)T \)
31 \( 1 + (-0.639 - 0.768i)T \)
37 \( 1 + (-0.470 + 0.882i)T \)
41 \( 1 + (-0.755 + 0.654i)T \)
43 \( 1 + (-0.639 + 0.768i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (0.523 + 0.852i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (-0.891 + 0.452i)T \)
67 \( 1 + (0.996 - 0.0815i)T \)
71 \( 1 + (0.933 + 0.359i)T \)
73 \( 1 + (-0.162 - 0.986i)T \)
79 \( 1 + (-0.699 + 0.714i)T \)
83 \( 1 + (0.999 + 0.0407i)T \)
89 \( 1 + (-0.999 - 0.0203i)T \)
97 \( 1 + (0.983 + 0.182i)T \)
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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

−22.36926603921513524210786206278, −21.49400637032707093011209425805, −20.47427458486645642429719094722, −19.8840325293516105358372191247, −19.24233107614159617225986621623, −18.36866759177780365152699826036, −17.244507382470793010588118201202, −16.53786091056950362993303941684, −15.639072402541160998826593539292, −14.37124899122237674977761481510, −13.66914467308604979031257373179, −12.94191698000593527269324218198, −12.32625781566326422723456348822, −11.8045399304571288351645463253, −10.55753222131995645158286183604, −9.59200591679415418586791321377, −8.8189686536113297048860792825, −7.131431883487133761162381939587, −6.79438331947939668086064145254, −5.604707283466434220688041177847, −4.83697905910234387543238940828, −3.80624934693357486614656462891, −2.40503613694750831739083679445, −1.71749973501148059513939263912, −0.40610550347948274075615723747, 2.51701693072238623182011934090, 3.32457222437760672933468075437, 3.87303243109167712217156426758, 5.2956573027773407375851288982, 6.05388122389255861640742081151, 6.48143704682543045152826190372, 7.88750251389957936260546006238, 8.80491767958000647861658671215, 10.01835701495183908592505607581, 10.678330078743926886031787222599, 11.569032444121401898571508500, 12.59706244624158730832645500374, 13.45190639425973635738681289553, 14.45015233704225489284208523296, 15.131134981487647807638591061442, 15.59889318846904479174471696679, 16.74153815543556363290229409247, 17.0620632636104603000498491396, 18.29941021981208957684191844171, 19.30436031403067820275753609130, 20.26947932853757652887587997393, 21.43119119137624041663641734153, 21.78782553842441229163053822730, 22.444332272087818986125531743809, 23.04597235245494818542664695205

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