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Properties

Label 1-167-167.48-r0-0-0
Degree $1$
Conductor $167$
Sign $0.130 - 0.991i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (0.280 + 0.959i)2-s + (0.489 − 0.872i)3-s + (−0.843 + 0.537i)4-s + (−0.982 − 0.188i)5-s + (0.974 + 0.225i)6-s + (−0.942 − 0.334i)7-s + (−0.752 − 0.658i)8-s + (−0.521 − 0.853i)9-s + (−0.0944 − 0.995i)10-s + (0.553 − 0.832i)11-s + (0.0567 + 0.998i)12-s + (−0.800 − 0.599i)13-s + (0.0567 − 0.998i)14-s + (−0.644 + 0.764i)15-s + (0.421 − 0.906i)16-s + (0.822 + 0.569i)17-s + ⋯
L(s)  = 1  + (0.280 + 0.959i)2-s + (0.489 − 0.872i)3-s + (−0.843 + 0.537i)4-s + (−0.982 − 0.188i)5-s + (0.974 + 0.225i)6-s + (−0.942 − 0.334i)7-s + (−0.752 − 0.658i)8-s + (−0.521 − 0.853i)9-s + (−0.0944 − 0.995i)10-s + (0.553 − 0.832i)11-s + (0.0567 + 0.998i)12-s + (−0.800 − 0.599i)13-s + (0.0567 − 0.998i)14-s + (−0.644 + 0.764i)15-s + (0.421 − 0.906i)16-s + (0.822 + 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5260954795 - 0.4612457180i\)
\(L(\frac12)\) \(\approx\) \(0.5260954795 - 0.4612457180i\)
\(L(1)\) \(\approx\) \(0.8407676001 - 0.06763438931i\)
\(L(1)\) \(\approx\) \(0.8407676001 - 0.06763438931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.280 + 0.959i)T \)
3 \( 1 + (0.489 - 0.872i)T \)
5 \( 1 + (-0.982 - 0.188i)T \)
7 \( 1 + (-0.942 - 0.334i)T \)
11 \( 1 + (0.553 - 0.832i)T \)
13 \( 1 + (-0.800 - 0.599i)T \)
17 \( 1 + (0.822 + 0.569i)T \)
19 \( 1 + (-0.0189 - 0.999i)T \)
23 \( 1 + (-0.993 + 0.113i)T \)
29 \( 1 + (-0.993 - 0.113i)T \)
31 \( 1 + (-0.243 - 0.969i)T \)
37 \( 1 + (-0.521 + 0.853i)T \)
41 \( 1 + (-0.881 + 0.472i)T \)
43 \( 1 + (0.726 + 0.686i)T \)
47 \( 1 + (0.988 - 0.150i)T \)
53 \( 1 + (0.206 - 0.978i)T \)
59 \( 1 + (0.822 - 0.569i)T \)
61 \( 1 + (0.997 - 0.0756i)T \)
67 \( 1 + (-0.982 + 0.188i)T \)
71 \( 1 + (0.614 - 0.788i)T \)
73 \( 1 + (0.421 + 0.906i)T \)
79 \( 1 + (-0.965 + 0.261i)T \)
83 \( 1 + (0.280 - 0.959i)T \)
89 \( 1 + (-0.644 - 0.764i)T \)
97 \( 1 + (-0.243 + 0.969i)T \)
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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

−27.82091529983777800860870786212, −27.13185494825964839870039607332, −26.24220764190964987110424191380, −25.1228458104246321306694798664, −23.612554275934410486483191113031, −22.51187376410772050406190607323, −22.2427083301023324683859862166, −20.91534829411169249053652666056, −20.0380117023208173819382102196, −19.37592718383212389170842611741, −18.57478771650282764738127136135, −16.82682491098914271546397263080, −15.794501261713419226227269612102, −14.74353679022470794835757390255, −14.077717618528678845571154859750, −12.3868988612171274918025837446, −11.92536406980248693555506437234, −10.48183191572644469201522421072, −9.69075366581546006478396693708, −8.81556972743739063418173098974, −7.32251632510639486599397893423, −5.47270562065852226478096883294, −4.15115399609145401085799531322, −3.510601443308983112184704538, −2.25386443173062606162332897480, 0.49056005594067984141557686469, 3.11384370564357532993773733397, 3.93943958804957610750221177831, 5.705374165688080406977975567330, 6.813894858158242143234514960890, 7.67128246113927498434317366542, 8.51855952082315008714582886126, 9.676117048465397855566164298028, 11.68845105036307114840523415167, 12.65361845265635517824398303954, 13.39734958624456899800311309685, 14.53126560146817347251642653932, 15.39662966636125655805925207443, 16.519161396901237087496392012548, 17.32331471021465920501722440990, 18.75935632287649458810461432321, 19.37324661064663146649918868331, 20.31039160158597308740663811038, 22.03551269536861129827046002043, 22.81268732198756498856717369029, 23.972926187474258398953248523425, 24.20666723365687369450451710473, 25.47308143811786973981410068018, 26.20281368429141268580917325548, 27.08738873946050977386726501450

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