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Properties

Label 1-667-667.108-r1-0-0
Degree $1$
Conductor $667$
Sign $-0.947 - 0.320i$
Analytic cond. $71.6791$
Root an. cond. $71.6791$
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  + (−0.505 + 0.862i)2-s + (0.607 − 0.794i)3-s + (−0.488 − 0.872i)4-s + (−0.182 − 0.983i)5-s + (0.377 + 0.925i)6-s + (0.339 + 0.940i)7-s + (0.999 + 0.0203i)8-s + (−0.262 − 0.965i)9-s + (0.940 + 0.339i)10-s + (−0.953 + 0.301i)11-s + (−0.989 − 0.142i)12-s + (0.992 + 0.122i)13-s + (−0.983 − 0.182i)14-s + (−0.891 − 0.452i)15-s + (−0.523 + 0.852i)16-s + (−0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.505 + 0.862i)2-s + (0.607 − 0.794i)3-s + (−0.488 − 0.872i)4-s + (−0.182 − 0.983i)5-s + (0.377 + 0.925i)6-s + (0.339 + 0.940i)7-s + (0.999 + 0.0203i)8-s + (−0.262 − 0.965i)9-s + (0.940 + 0.339i)10-s + (−0.953 + 0.301i)11-s + (−0.989 − 0.142i)12-s + (0.992 + 0.122i)13-s + (−0.983 − 0.182i)14-s + (−0.891 − 0.452i)15-s + (−0.523 + 0.852i)16-s + (−0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(71.6791\)
Root analytic conductor: \(71.6791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (1:\ ),\ -0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07728246545 - 0.4702634957i\)
\(L(\frac12)\) \(\approx\) \(0.07728246545 - 0.4702634957i\)
\(L(1)\) \(\approx\) \(0.8154049316 - 0.05503745205i\)
\(L(1)\) \(\approx\) \(0.8154049316 - 0.05503745205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.505 + 0.862i)T \)
3 \( 1 + (0.607 - 0.794i)T \)
5 \( 1 + (-0.182 - 0.983i)T \)
7 \( 1 + (0.339 + 0.940i)T \)
11 \( 1 + (-0.953 + 0.301i)T \)
13 \( 1 + (0.992 + 0.122i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.872 - 0.488i)T \)
31 \( 1 + (0.242 - 0.970i)T \)
37 \( 1 + (-0.965 + 0.262i)T \)
41 \( 1 + (-0.755 - 0.654i)T \)
43 \( 1 + (-0.242 - 0.970i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (0.101 - 0.994i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.202 + 0.979i)T \)
67 \( 1 + (0.301 - 0.953i)T \)
71 \( 1 + (-0.557 - 0.830i)T \)
73 \( 1 + (0.574 - 0.818i)T \)
79 \( 1 + (-0.852 + 0.523i)T \)
83 \( 1 + (-0.591 + 0.806i)T \)
89 \( 1 + (-0.891 + 0.452i)T \)
97 \( 1 + (-0.470 + 0.882i)T \)
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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

−22.84312460165772508206532304101, −21.89568356775152669852381926969, −21.08470092939173301908757657159, −20.509027584992188087543478352393, −19.87377326250365611314180768684, −18.86642375330683561035149433932, −18.25897657815258071845851826511, −17.42667016237889429001295083189, −16.11830109941633777476957737196, −15.839873473793078221391321035710, −14.357140648648412813065637107180, −13.85587888805730987395189696871, −13.1244948505548291145255017019, −11.53966348923708231897344469320, −10.99925917613292435936447041026, −10.31730484618342350056265141374, −9.71692519477998957010820044267, −8.473975624695301559717448167301, −7.84983377518684305647571421271, −6.99024171571244813288821446794, −5.256108540001110020646355538943, −4.2069172501048298279147778311, −3.301880439809983502731027640045, −2.80500609700605634750560260973, −1.41048301838245901217087961992, 0.12413028849362840324448658526, 1.350038981166207554536286570699, 2.177609687758546058940001671129, 3.79775377861959657346065638501, 5.1249761518783164630264434638, 5.76433385382839958794782948900, 6.846354824355539308764315820619, 7.897584091917627586649726435828, 8.484923789736655507634981690772, 8.950359219240406192828224724627, 10.00586282396857834188400230642, 11.37130578641152043395233290307, 12.356509419576306850852874492305, 13.28731182352384306878019720892, 13.74810478893581655134753990570, 15.20362405958658357878332460739, 15.35783894078903644018146266977, 16.385891023064135698232616561555, 17.48014456506053660388588204114, 18.09047582655068721802901109490, 18.780887955195256768382063432848, 19.54624687646471516226548006240, 20.483988088038162730253109745789, 21.10540544662221269557061301302, 22.47579780049240951917022226085

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