Properties

Label 1-179-179.32-r1-0-0
Degree $1$
Conductor $179$
Sign $0.998 + 0.0549i$
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  + (0.984 + 0.175i)2-s + (0.977 + 0.210i)3-s + (0.938 + 0.345i)4-s + (0.713 − 0.700i)5-s + (0.925 + 0.378i)6-s + (0.329 − 0.944i)7-s + (0.863 + 0.505i)8-s + (0.911 + 0.411i)9-s + (0.825 − 0.564i)10-s + (−0.880 + 0.474i)11-s + (0.844 + 0.535i)12-s + (0.295 + 0.955i)13-s + (0.489 − 0.871i)14-s + (0.844 − 0.535i)15-s + (0.760 + 0.648i)16-s + (−0.520 − 0.854i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.175i)2-s + (0.977 + 0.210i)3-s + (0.938 + 0.345i)4-s + (0.713 − 0.700i)5-s + (0.925 + 0.378i)6-s + (0.329 − 0.944i)7-s + (0.863 + 0.505i)8-s + (0.911 + 0.411i)9-s + (0.825 − 0.564i)10-s + (−0.880 + 0.474i)11-s + (0.844 + 0.535i)12-s + (0.295 + 0.955i)13-s + (0.489 − 0.871i)14-s + (0.844 − 0.535i)15-s + (0.760 + 0.648i)16-s + (−0.520 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.998 + 0.0549i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ 0.998 + 0.0549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.519892366 + 0.1517457671i\)
\(L(\frac12)\) \(\approx\) \(5.519892366 + 0.1517457671i\)
\(L(1)\) \(\approx\) \(2.949520239 + 0.1246614375i\)
\(L(1)\) \(\approx\) \(2.949520239 + 0.1246614375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.984 + 0.175i)T \)
3 \( 1 + (0.977 + 0.210i)T \)
5 \( 1 + (0.713 - 0.700i)T \)
7 \( 1 + (0.329 - 0.944i)T \)
11 \( 1 + (-0.880 + 0.474i)T \)
13 \( 1 + (0.295 + 0.955i)T \)
17 \( 1 + (-0.520 - 0.854i)T \)
19 \( 1 + (-0.994 - 0.105i)T \)
23 \( 1 + (0.261 - 0.965i)T \)
29 \( 1 + (-0.394 + 0.918i)T \)
31 \( 1 + (-0.0529 - 0.998i)T \)
37 \( 1 + (0.394 + 0.918i)T \)
41 \( 1 + (-0.607 + 0.794i)T \)
43 \( 1 + (0.0176 + 0.999i)T \)
47 \( 1 + (0.997 + 0.0705i)T \)
53 \( 1 + (-0.804 + 0.593i)T \)
59 \( 1 + (-0.999 + 0.0352i)T \)
61 \( 1 + (0.990 + 0.140i)T \)
67 \( 1 + (-0.863 + 0.505i)T \)
71 \( 1 + (-0.550 - 0.835i)T \)
73 \( 1 + (0.896 - 0.442i)T \)
79 \( 1 + (-0.362 + 0.932i)T \)
83 \( 1 + (-0.458 - 0.888i)T \)
89 \( 1 + (-0.984 + 0.175i)T \)
97 \( 1 + (0.737 - 0.675i)T \)
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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

−26.90012802497720734922962771395, −25.61909200792548977805903825881, −25.31907116947170446660548897370, −24.27909320489602823458663218591, −23.31995377261451402813240499078, −21.97470899642150359551316658124, −21.39483257146648139406978803460, −20.68613843628545785131280633913, −19.37224548312150986479260831854, −18.65803821826544196700358181131, −17.52960485373791981738364366787, −15.54171028037104059315316458168, −15.223817795466343827838719757205, −14.18243357578845423001112185573, −13.26536662659155551025853916711, −12.59648936691693605333319125986, −11.02642499065187219479103261865, −10.19938122190387069828812596048, −8.726239666819538098589708415715, −7.57955780061877391670447376727, −6.23595068125266511245475002325, −5.36869934634642760445521328398, −3.65750110330896216278676868242, −2.6292999263190586830209752110, −1.85715012804740420911251925801, 1.63979508053398282339104987273, 2.69552291758895121581060238777, 4.35064539333480664043181914455, 4.76824802515833921399628251375, 6.51170413046829814534899778724, 7.58932457402364671996432418507, 8.717568612114917028236293992926, 10.010788982680045199300958629250, 11.07534251051810419832126772988, 12.730121648953236900864854842948, 13.37898396339313865996932613008, 14.10331939754720256172161734613, 15.05466811049611860456465072513, 16.234079357868644049061644576562, 16.94220746759505066747959429841, 18.43043171848081014247999654919, 19.94934899919073258417980037222, 20.64463421086228326974330654512, 21.0808407860040055990519738309, 22.15062453126828932019561753142, 23.581425326370218625996457700, 24.14244142194580779472783677396, 25.18870386757140402786876318016, 25.93031687794669572208251621492, 26.725867257261710592913858633231

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