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Properties

Label 1-319-319.112-r1-0-0
Degree $1$
Conductor $319$
Sign $0.417 - 0.908i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (−0.473 − 0.880i)2-s + (−0.550 − 0.834i)3-s + (−0.550 + 0.834i)4-s + (0.983 − 0.178i)5-s + (−0.473 + 0.880i)6-s + (−0.936 − 0.351i)7-s + (0.995 + 0.0896i)8-s + (−0.393 + 0.919i)9-s + (−0.623 − 0.781i)10-s + 12-s + (−0.753 + 0.657i)13-s + (0.134 + 0.990i)14-s + (−0.691 − 0.722i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + (0.995 − 0.0896i)18-s + ⋯
L(s)  = 1  + (−0.473 − 0.880i)2-s + (−0.550 − 0.834i)3-s + (−0.550 + 0.834i)4-s + (0.983 − 0.178i)5-s + (−0.473 + 0.880i)6-s + (−0.936 − 0.351i)7-s + (0.995 + 0.0896i)8-s + (−0.393 + 0.919i)9-s + (−0.623 − 0.781i)10-s + 12-s + (−0.753 + 0.657i)13-s + (0.134 + 0.990i)14-s + (−0.691 − 0.722i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + (0.995 − 0.0896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8226586349 - 0.5273578598i\)
\(L(\frac12)\) \(\approx\) \(0.8226586349 - 0.5273578598i\)
\(L(1)\) \(\approx\) \(0.5945074209 - 0.3748507076i\)
\(L(1)\) \(\approx\) \(0.5945074209 - 0.3748507076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.473 - 0.880i)T \)
3 \( 1 + (-0.550 - 0.834i)T \)
5 \( 1 + (0.983 - 0.178i)T \)
7 \( 1 + (-0.936 - 0.351i)T \)
13 \( 1 + (-0.753 + 0.657i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.936 + 0.351i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.473 + 0.880i)T \)
37 \( 1 + (0.858 - 0.512i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.858 + 0.512i)T \)
53 \( 1 + (0.473 + 0.880i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.0448 - 0.998i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (0.691 + 0.722i)T \)
79 \( 1 + (0.393 - 0.919i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.0448 - 0.998i)T \)
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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

−25.44629564214231184286343051952, −24.30369842767444653481477472835, −23.276090131328638739604952638120, −22.291945613536395580859355161850, −21.97734824555659239552025089849, −20.732242346888745237002276521788, −19.520377322277551464385331595210, −18.555270647823050012677578756136, −17.54469248729088681977718596311, −16.97170903605609545151331834666, −16.19280807431269036162726031806, −15.15996346022741787617673325763, −14.58681405910548271828443743503, −13.31266418491100620118583916496, −12.30689407700036543445340228272, −10.65514004056142362198105556658, −10.00997773876772264823856410945, −9.446168689915077504438333762878, −8.33439984287540955825619965979, −6.804214395406316419686630491734, −5.97505125072136105985470211816, −5.404941926712173162687724451385, −4.08571204654439173725755583275, −2.48903318784769837773844736647, −0.54921235985428212247746626774, 0.7478863562441804020204789011, 1.90270322263097189246966862717, 2.85190852777865354093087788527, 4.44590289704954460911746582533, 5.75758764303471875660061855047, 6.80501848227620061743924667278, 7.80433239505084363490439645263, 9.14820282056464673649356154658, 9.95528890291513905596958309009, 10.75730662168591534771857745285, 12.09765593504276752335366056185, 12.55508321491411779539916871250, 13.56541461294498402918639730708, 14.15585967893120813188731698281, 16.35779198240461078209771913237, 16.85205914827981101986950259303, 17.669020844697776698926561060429, 18.55318041300454534331840953742, 19.301682948720463169864502791673, 20.07718604347104655105497929979, 21.29017512747549016069865612146, 21.98372446752158296935724343532, 22.84841212127843798051716440449, 23.71458228675677402529819187318, 25.06769632145645807549375957134

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