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Properties

Label 1-667-667.151-r0-0-0
Degree $1$
Conductor $667$
Sign $0.402 + 0.915i$
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  + (−0.996 + 0.0815i)2-s + (−0.0203 + 0.999i)3-s + (0.986 − 0.162i)4-s + (−0.591 + 0.806i)5-s + (−0.0611 − 0.998i)6-s + (−0.523 − 0.852i)7-s + (−0.970 + 0.242i)8-s + (−0.999 − 0.0407i)9-s + (0.523 − 0.852i)10-s + (0.862 + 0.505i)11-s + (0.142 + 0.989i)12-s + (0.101 − 0.994i)13-s + (0.591 + 0.806i)14-s + (−0.794 − 0.607i)15-s + (0.947 − 0.320i)16-s + (0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0815i)2-s + (−0.0203 + 0.999i)3-s + (0.986 − 0.162i)4-s + (−0.591 + 0.806i)5-s + (−0.0611 − 0.998i)6-s + (−0.523 − 0.852i)7-s + (−0.970 + 0.242i)8-s + (−0.999 − 0.0407i)9-s + (0.523 − 0.852i)10-s + (0.862 + 0.505i)11-s + (0.142 + 0.989i)12-s + (0.101 − 0.994i)13-s + (0.591 + 0.806i)14-s + (−0.794 − 0.607i)15-s + (0.947 − 0.320i)16-s + (0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5954223794 + 0.3884219927i\)
\(L(\frac12)\) \(\approx\) \(0.5954223794 + 0.3884219927i\)
\(L(1)\) \(\approx\) \(0.5873599770 + 0.2282772377i\)
\(L(1)\) \(\approx\) \(0.5873599770 + 0.2282772377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.996 + 0.0815i)T \)
3 \( 1 + (-0.0203 + 0.999i)T \)
5 \( 1 + (-0.591 + 0.806i)T \)
7 \( 1 + (-0.523 - 0.852i)T \)
11 \( 1 + (0.862 + 0.505i)T \)
13 \( 1 + (0.101 - 0.994i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (-0.986 + 0.162i)T \)
31 \( 1 + (0.979 + 0.202i)T \)
37 \( 1 + (0.999 + 0.0407i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.979 - 0.202i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.339 - 0.940i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.768 - 0.639i)T \)
67 \( 1 + (-0.862 + 0.505i)T \)
71 \( 1 + (0.685 + 0.728i)T \)
73 \( 1 + (-0.488 + 0.872i)T \)
79 \( 1 + (-0.947 - 0.320i)T \)
83 \( 1 + (0.262 - 0.965i)T \)
89 \( 1 + (-0.794 + 0.607i)T \)
97 \( 1 + (-0.917 - 0.396i)T \)
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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

−22.8151260780650536271029487757, −21.55818040777516991912712282766, −20.86124008285076699579302682833, −19.630993402161375676272959939514, −19.29088424442839517594065930931, −18.8336169340955642947552320467, −17.7931863894833823569136588322, −16.71275114943628635475148540846, −16.5606568607625499391305967535, −15.36163161944885308762500900919, −14.43766980593079003084340177609, −13.162943900454909586451557494644, −12.23745802203192487881944628529, −11.89734634417968961066368297121, −11.06366416881593160749163894934, −9.57849940648315778249720155629, −8.80983976146885257163365142175, −8.35135122616178399499772293801, −7.32255999501678872366114403658, −6.36797563331279097486001101819, −5.73923499945829990210671824336, −4.0095578020307597196342759660, −2.79570130017801123915658664626, −1.7325272518587227446245553493, −0.7448821498959429690494614394, 0.806678177816924222734986097857, 2.657560578085987629588214950063, 3.45600769822474346554446083181, 4.34662746654625287486794779636, 5.90353092577675900246475855326, 6.70287897019133734435874597075, 7.67287674093068888147055296881, 8.44077803559826078209176537616, 9.75365221371826156382266186016, 10.05913561833161968466633555747, 10.91518988064788294926433616016, 11.596906748458385582770713153838, 12.68339529966561701771163171239, 14.367629149571944986226863430156, 14.76196039252258427285657199726, 15.71432454594333211783090154806, 16.329486681827843726864337224587, 17.19770893944575294697631373056, 17.79440353600035288029476788958, 19.07755014223598747010587624763, 19.59965789590059607510682408101, 20.31769579532623397451282226094, 21.05839362642273036448120245050, 22.19434390200133725410339393661, 22.97922776582670235466779464147

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