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Properties

Label 1-667-667.4-r0-0-0
Degree $1$
Conductor $667$
Sign $0.942 - 0.334i$
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  + (−0.917 + 0.396i)2-s + (−0.101 + 0.994i)3-s + (0.685 − 0.728i)4-s + (0.0203 − 0.999i)5-s + (−0.301 − 0.953i)6-s + (−0.377 + 0.925i)7-s + (−0.339 + 0.940i)8-s + (−0.979 − 0.202i)9-s + (0.377 + 0.925i)10-s + (−0.882 + 0.470i)11-s + (0.654 + 0.755i)12-s + (0.488 − 0.872i)13-s + (−0.0203 − 0.999i)14-s + (0.992 + 0.122i)15-s + (−0.0611 − 0.998i)16-s + (0.142 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.917 + 0.396i)2-s + (−0.101 + 0.994i)3-s + (0.685 − 0.728i)4-s + (0.0203 − 0.999i)5-s + (−0.301 − 0.953i)6-s + (−0.377 + 0.925i)7-s + (−0.339 + 0.940i)8-s + (−0.979 − 0.202i)9-s + (0.377 + 0.925i)10-s + (−0.882 + 0.470i)11-s + (0.654 + 0.755i)12-s + (0.488 − 0.872i)13-s + (−0.0203 − 0.999i)14-s + (0.992 + 0.122i)15-s + (−0.0611 − 0.998i)16-s + (0.142 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5383799767 - 0.09280270700i\)
\(L(\frac12)\) \(\approx\) \(0.5383799767 - 0.09280270700i\)
\(L(1)\) \(\approx\) \(0.5643617280 + 0.1376706514i\)
\(L(1)\) \(\approx\) \(0.5643617280 + 0.1376706514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.917 + 0.396i)T \)
3 \( 1 + (-0.101 + 0.994i)T \)
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

−23.03250973415144244639294723630, −21.841649688308614392461087338608, −21.150328244471080478969376800969, −19.99789756648635552305096374361, −19.32285506913449071685584570186, −18.78369773515058981244818020916, −18.11611221538201127170352816155, −17.247539428947256198838377880247, −16.617691171120478097366453371544, −15.5527779310850655686270497276, −14.41433367607148569220682269629, −13.372958226010726128665818062013, −12.92144291226619460252640334147, −11.60982028590535400204983236632, −10.97538904547022887928948898720, −10.40723154065817041583994978240, −9.26297898296079618600661351770, −8.07605104306697857599971778918, −7.578193232034026476809398071419, −6.52271045741044621995884940565, −6.195449688970711917935375604101, −4.090640906551718781632411034085, −3.00495300383738971446021947366, −2.22697922439883449018840929961, −1.011864784757958693320784002433, 0.429899321837098867030248080598, 2.10755847927619947696804010097, 3.17306521328533469204740389424, 4.7316902203556233906813488317, 5.46304532048414735556809252687, 6.08340540803690651629939898760, 7.62585774010459931172402761100, 8.55661832055666662558178990200, 9.022416226397078274141540435066, 9.99393756058830378151910700565, 10.524098184372669435758079398, 11.75768528534612797147839426638, 12.459790751929956342602325067, 13.70501265461896685372767403676, 15.01775102123473271644680235594, 15.54566840613167266209662406978, 16.12526668322555599487781092808, 16.81456834408232282199926821323, 17.76319876357243735333439076659, 18.45033817577712318865698792173, 19.51691651121335346399086608584, 20.49932704993666146489641561348, 20.75956470819911892976702938007, 21.72339138554012534205860117357, 22.9775649182978853648942819895

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