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Properties

Label 1-181-181.33-r0-0-0
Degree $1$
Conductor $181$
Sign $-0.962 + 0.271i$
Analytic cond. $0.840560$
Root an. cond. $0.840560$
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  + (−0.559 − 0.829i)2-s + (−0.241 − 0.970i)3-s + (−0.374 + 0.927i)4-s + (−0.104 + 0.994i)5-s + (−0.669 + 0.743i)6-s + (0.5 − 0.866i)7-s + (0.978 − 0.207i)8-s + (−0.882 + 0.469i)9-s + (0.882 − 0.469i)10-s + (−0.615 − 0.788i)11-s + (0.990 + 0.139i)12-s + (−0.997 − 0.0697i)13-s + (−0.997 + 0.0697i)14-s + (0.990 − 0.139i)15-s + (−0.719 − 0.694i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (−0.241 − 0.970i)3-s + (−0.374 + 0.927i)4-s + (−0.104 + 0.994i)5-s + (−0.669 + 0.743i)6-s + (0.5 − 0.866i)7-s + (0.978 − 0.207i)8-s + (−0.882 + 0.469i)9-s + (0.882 − 0.469i)10-s + (−0.615 − 0.788i)11-s + (0.990 + 0.139i)12-s + (−0.997 − 0.0697i)13-s + (−0.997 + 0.0697i)14-s + (0.990 − 0.139i)15-s + (−0.719 − 0.694i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $-0.962 + 0.271i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ -0.962 + 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05552470774 - 0.4017334057i\)
\(L(\frac12)\) \(\approx\) \(-0.05552470774 - 0.4017334057i\)
\(L(1)\) \(\approx\) \(0.4141657472 - 0.3858925180i\)
\(L(1)\) \(\approx\) \(0.4141657472 - 0.3858925180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T \)
3 \( 1 + (-0.241 - 0.970i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (-0.997 - 0.0697i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.0348 - 0.999i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.961 + 0.275i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.990 - 0.139i)T \)
53 \( 1 + (-0.848 - 0.529i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.848 + 0.529i)T \)
83 \( 1 + (-0.438 + 0.898i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.615 - 0.788i)T \)
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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

−27.85457431312188537368796498061, −26.990381989416630029810054519744, −25.88853138054448317809215519849, −25.111215657802771018362033437042, −24.03631565648178543989292850232, −23.33491647409869153277096014526, −22.0295078594370444918829276119, −21.143320357630311412602412172704, −20.0867531259094956896740169170, −19.08058433557410045857109470349, −17.48890306232012400843939742781, −17.272636207187520498790664999313, −16.01737834524308976447396493948, −15.25627658641404868859831240851, −14.65785598593217518916603752051, −12.96652032023507587644851604132, −11.79077049254677401624756709119, −10.455087904588524879286813352530, −9.481573299246514519070513299144, −8.68258642887668175658840004767, −7.71930728799127896859150072663, −5.96329387006816314154923247822, −5.089566965475033427018344608718, −4.35398697764994804635975937071, −1.98191453125568904198541207860, 0.38114885926058549153719060563, 2.131052898459122661086986284828, 3.052199296941166657682734939, 4.700101952671038328836884594601, 6.54192171550371348953243022914, 7.54462047967065892307174099620, 8.21862126063355264030850327192, 9.92236902759609265038614130080, 10.986438739329138174146488435224, 11.4585163673555255566213442603, 12.78790264954373930187686778649, 13.69087832628383657052894432530, 14.61059535087578737870748450232, 16.50994759805490679178874074554, 17.35588524304036560700624542427, 18.27640936509405184518647661028, 18.9119547392348758352846825614, 19.78583321880313555533545385940, 20.80820898578718831068347881055, 22.04634727742529371515238572895, 22.83878652066434651168149744308, 23.815608546630717583091945392921, 24.93295857090684303299060827970, 26.19823738412603070470152231101, 26.757939128481824707886112675248

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