Properties

Label 1-176-176.157-r0-0-0
Degree 11
Conductor 176176
Sign 0.8390.542i0.839 - 0.542i
Analytic cond. 0.8173400.817340
Root an. cond. 0.8173400.817340
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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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

Λ(s)=(176s/2ΓR(s)L(s)=((0.8390.542i)Λ(1s)\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}
Λ(s)=(176s/2ΓR(s)L(s)=((0.8390.542i)Λ(1s)\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: 11
Conductor: 176176    =    24112^{4} \cdot 11
Sign: 0.8390.542i0.839 - 0.542i
Analytic conductor: 0.8173400.817340
Root analytic conductor: 0.8173400.817340
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ176(157,)\chi_{176} (157, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 176, (0: ), 0.8390.542i)(1,\ 176,\ (0:\ ),\ 0.839 - 0.542i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.1513596170.3395421650i1.151359617 - 0.3395421650i
L(12)L(\frac12) \approx 1.1513596170.3395421650i1.151359617 - 0.3395421650i
L(1)L(1) \approx 1.0842696410.2229968010i1.084269641 - 0.2229968010i
L(1)L(1) \approx 1.0842696410.2229968010i1.084269641 - 0.2229968010i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
11 1 1
good3 1+(0.5870.809i)T 1 + (-0.587 - 0.809i)T
5 1+(0.9510.309i)T 1 + (0.951 - 0.309i)T
7 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
13 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
17 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
19 1+(0.5870.809i)T 1 + (-0.587 - 0.809i)T
23 1T 1 - T
29 1+(0.5870.809i)T 1 + (0.587 - 0.809i)T
31 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
37 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
41 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
43 1iT 1 - iT
47 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
53 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
59 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
61 1+(0.951+0.309i)T 1 + (-0.951 + 0.309i)T
67 1+iT 1 + iT
71 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
73 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
79 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
83 1+(0.951+0.309i)T 1 + (-0.951 + 0.309i)T
89 1T 1 - T
97 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line