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Properties

Label 1-179-179.2-r1-0-0
Degree $1$
Conductor $179$
Sign $0.224 - 0.974i$
Analytic cond. $19.2362$
Root an. cond. $19.2362$
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  + (0.999 + 0.0352i)2-s + (−0.783 − 0.621i)3-s + (0.997 + 0.0705i)4-s + (0.158 − 0.987i)5-s + (−0.760 − 0.648i)6-s + (0.969 − 0.244i)7-s + (0.994 + 0.105i)8-s + (0.227 + 0.973i)9-s + (0.192 − 0.981i)10-s + (0.863 + 0.505i)11-s + (−0.737 − 0.675i)12-s + (−0.635 − 0.772i)13-s + (0.977 − 0.210i)14-s + (−0.737 + 0.675i)15-s + (0.990 + 0.140i)16-s + (0.911 − 0.411i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0352i)2-s + (−0.783 − 0.621i)3-s + (0.997 + 0.0705i)4-s + (0.158 − 0.987i)5-s + (−0.760 − 0.648i)6-s + (0.969 − 0.244i)7-s + (0.994 + 0.105i)8-s + (0.227 + 0.973i)9-s + (0.192 − 0.981i)10-s + (0.863 + 0.505i)11-s + (−0.737 − 0.675i)12-s + (−0.635 − 0.772i)13-s + (0.977 − 0.210i)14-s + (−0.737 + 0.675i)15-s + (0.990 + 0.140i)16-s + (0.911 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.577317267 - 2.051733676i\)
\(L(\frac12)\) \(\approx\) \(2.577317267 - 2.051733676i\)
\(L(1)\) \(\approx\) \(1.761549807 - 0.6995739915i\)
\(L(1)\) \(\approx\) \(1.761549807 - 0.6995739915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.999 + 0.0352i)T \)
3 \( 1 + (-0.783 - 0.621i)T \)
5 \( 1 + (0.158 - 0.987i)T \)
7 \( 1 + (0.969 - 0.244i)T \)
11 \( 1 + (0.863 + 0.505i)T \)
13 \( 1 + (-0.635 - 0.772i)T \)
17 \( 1 + (0.911 - 0.411i)T \)
19 \( 1 + (-0.329 + 0.944i)T \)
23 \( 1 + (0.0529 - 0.998i)T \)
29 \( 1 + (-0.520 - 0.854i)T \)
31 \( 1 + (-0.579 + 0.815i)T \)
37 \( 1 + (0.520 - 0.854i)T \)
41 \( 1 + (0.688 - 0.725i)T \)
43 \( 1 + (-0.949 + 0.312i)T \)
47 \( 1 + (0.295 + 0.955i)T \)
53 \( 1 + (-0.427 - 0.904i)T \)
59 \( 1 + (0.804 - 0.593i)T \)
61 \( 1 + (-0.825 + 0.564i)T \)
67 \( 1 + (-0.994 + 0.105i)T \)
71 \( 1 + (-0.489 + 0.871i)T \)
73 \( 1 + (0.394 + 0.918i)T \)
79 \( 1 + (0.925 + 0.378i)T \)
83 \( 1 + (0.662 + 0.749i)T \)
89 \( 1 + (-0.999 + 0.0352i)T \)
97 \( 1 + (-0.713 + 0.700i)T \)
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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

−27.41386902511171461344223428622, −26.38365617742983634787634946033, −25.291753186687283428787801626608, −23.98478128030494783871868759522, −23.52051969892130977946642269277, −22.17394736841059243807891893360, −21.81312971013896557701183903453, −21.14257852166033729974772361288, −19.74311255191629020067094779791, −18.55719771789545742543897719416, −17.27238971286182029164556497478, −16.54328463506530358292373020691, −14.99482186917741742389729652280, −14.816534775956112067839114996713, −13.644122956972748270121742453260, −12.008128986183465674634284899056, −11.42892284128198576367400525669, −10.68050671487644367101649912355, −9.390492041242070669318054042447, −7.45454530295580115030934238692, −6.40814099272618054518247139919, −5.47045885834357447906819085228, −4.359307109406753832038607389862, −3.260316439976697678270417753901, −1.65593586595676466545440032832, 1.02806352965191872933749849666, 2.0663960844160521975476914502, 4.174657763598532475830147871355, 5.08578544587909498978890765284, 5.89257033178203912134660182284, 7.28422652616676221421655091973, 8.12815223924822923459532524723, 10.05515139142340785444036793568, 11.264064135231531095611524888561, 12.29369425545555065183705552771, 12.644507273000150067379330292190, 13.98938763798775703456593026106, 14.79303274481919996479686967653, 16.34138425106855428422905560275, 16.965155646938982378500560642690, 17.829618403143186666112040324775, 19.376742996273841427187425785641, 20.41126124789339799011807788832, 21.145549452944993487409383536316, 22.342187122274400170077555878760, 23.12420623486366087882387026589, 23.99121715928837163310245950118, 24.8845325570170458267038160213, 25.12751387984870151347380382378, 27.269540000689701633876939772268

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