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Properties

Label 1-319-319.169-r0-0-0
Degree $1$
Conductor $319$
Sign $0.944 + 0.329i$
Analytic cond. $1.48142$
Root an. cond. $1.48142$
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  + (−0.995 + 0.0896i)2-s + (0.983 + 0.178i)3-s + (0.983 − 0.178i)4-s + (0.858 + 0.512i)5-s + (−0.995 − 0.0896i)6-s + (0.473 − 0.880i)7-s + (−0.963 + 0.266i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + 12-s + (−0.550 + 0.834i)13-s + (−0.393 + 0.919i)14-s + (0.753 + 0.657i)15-s + (0.936 − 0.351i)16-s + (0.309 − 0.951i)17-s + (−0.963 − 0.266i)18-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.983 + 0.178i)3-s + (0.983 − 0.178i)4-s + (0.858 + 0.512i)5-s + (−0.995 − 0.0896i)6-s + (0.473 − 0.880i)7-s + (−0.963 + 0.266i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + 12-s + (−0.550 + 0.834i)13-s + (−0.393 + 0.919i)14-s + (0.753 + 0.657i)15-s + (0.936 − 0.351i)16-s + (0.309 − 0.951i)17-s + (−0.963 − 0.266i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(1.48142\)
Root analytic conductor: \(1.48142\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (0:\ ),\ 0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396700481 + 0.2365383534i\)
\(L(\frac12)\) \(\approx\) \(1.396700481 + 0.2365383534i\)
\(L(1)\) \(\approx\) \(1.149867152 + 0.1305880229i\)
\(L(1)\) \(\approx\) \(1.149867152 + 0.1305880229i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (0.983 + 0.178i)T \)
5 \( 1 + (0.858 + 0.512i)T \)
7 \( 1 + (0.473 - 0.880i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.473 + 0.880i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (-0.995 + 0.0896i)T \)
37 \( 1 + (-0.0448 + 0.998i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.0448 - 0.998i)T \)
53 \( 1 + (-0.995 + 0.0896i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.134 + 0.990i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (0.936 - 0.351i)T \)
73 \( 1 + (0.753 + 0.657i)T \)
79 \( 1 + (0.936 + 0.351i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.134 - 0.990i)T \)
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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

−25.3016217590687438044692002791, −24.55187755032580899137878666125, −23.9459391865250526973600517298, −21.765735641994392836775319469215, −21.47997005678744848862755112728, −20.37686154122825209456431024563, −19.79613229574075910696844761804, −18.79678840967849339426768925969, −17.87874753504573115518395852629, −17.375002241748960251949422288570, −16.01561507009324372277235619431, −15.17726774871349899441512686424, −14.35542897927129127473092806788, −12.98633567791526493000406326616, −12.37193035588176727213149548529, −11.0206649497037316016009999556, −9.75763597260754743398852804772, −9.28078603605351988052463695768, −8.31153918707000889593172860393, −7.61689104540458570394770028626, −6.23580726772379885958374688091, −5.13673343191472403190645370987, −3.25013411958641997846080051288, −2.22237919723525454571296753460, −1.40219991784540769690227806909, 1.486229296521182660425730720211, 2.3643750230742031439067178924, 3.56426224938052368867006258988, 5.123780905546805631578508619603, 6.738568211523039627253289222809, 7.34067873878773445073218512827, 8.37886539975332649891586465175, 9.47758557203473865080070022725, 10.05301496404914437774125212111, 10.88693356094751533676674676764, 12.18537672335958991174516047516, 13.77536346592574175897866253529, 14.25414668640404660036874389124, 15.11093133292522532752226535979, 16.447037566089139141502125345201, 16.99035385536220683869999633278, 18.342272338827547567451052024114, 18.64451714702623699981612560442, 19.90473653859569751954194904360, 20.59306748054923091898353000570, 21.21786718557113647446543771104, 22.32234345404203014497600816404, 23.83681271933273367741267522758, 24.64379447335672033374584371061, 25.41829670367451488721447803077

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