↑

Properties

Label 1-176-176.157-r0-0-0
Degree $1$
Conductor $176$
Sign $0.839 - 0.542i$
Analytic cond. $0.817340$
Root an. cond. $0.817340$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)3-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)9-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s − i·21-s − 23-s + (0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)35-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)9-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s − i·21-s − 23-s + (0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.839 - 0.542i$
Analytic conductor: \(0.817340\)
Root analytic conductor: \(0.817340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 176,\ (0:\ ),\ 0.839 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151359617 - 0.3395421650i\)
\(L(\frac12)\) \(\approx\) \(1.151359617 - 0.3395421650i\)
\(L(1)\) \(\approx\) \(1.084269641 - 0.2229968010i\)
\(L(1)\) \(\approx\) \(1.084269641 - 0.2229968010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 - T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.53766116384214648278770491828, −26.63481160141652424086714364427, −25.735337140789610488378139919764, −24.703206521472242390919196138850, −23.3557455506331838311502066103, −22.781378210222444238156059614612, −21.492320446833927255726090984464, −21.043902835908273261851905379031, −20.09979828793710112954424928421, −18.24927865911735701069932371454, −17.80748426119096913165566436473, −16.73674798406161690496259568202, −15.875966365327498919323041476098, −14.49602884570148604196307769126, −13.93282213756496101886290608733, −12.43958584494704134565795985785, −11.11525109557949386873667938864, −10.48510832949185301551499363211, −9.525511020457568469191294724515, −8.225695017695942845013601598725, −6.64084139726066347507210367907, −5.638905930232108566269704713717, −4.59922751846235182436760029962, −3.27212816759386222715352038933, −1.440029303350501005614341196252, 1.40812656019102209928066007356, 2.32072053764644670471483501257, 4.533016058229276416312892522881, 5.77825504122755555893440368289, 6.36649040833402979168270842281, 7.98095112394808131512495663725, 8.833822441347579555780186001670, 10.348505479521055386432365643441, 11.38309684347593311298478558890, 12.38124275024979873227964002965, 13.38107902475614448501657715952, 14.18081136126682837273637208607, 15.573276933928883731097155303100, 16.91765789223704557225602705576, 17.59403802620364570639247956183, 18.37120972538441019891119690228, 19.326211060996644877526739899937, 20.7831840498015471612109365440, 21.54696486537547593449692207487, 22.47997192748235887314778667869, 23.85241093583886843720719423597, 24.24604437553900076791816079446, 25.36233368121133173777389335120, 26.03223218163391975764473098355, 27.87876868826721167342094756663

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