Properties

Degree 1
Conductor 167
Sign $0.130 - 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.130 - 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (48, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.130 - 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5260954795 - 0.4612457180i$
$L(\frac12,\chi)$  $\approx$  $0.5260954795 - 0.4612457180i$
$L(\chi,1)$  $\approx$  0.8407676001 - 0.06763438931i
$L(1,\chi)$  $\approx$  0.8407676001 - 0.06763438931i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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