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Properties

Label 1-319-319.318-r1-0-0
Degree $1$
Conductor $319$
Sign $1$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s + 21-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s − 30-s − 31-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s − 13-s − 14-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s + 21-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s − 30-s − 31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $1$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{319} (318, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.252183559\)
\(L(\frac12)\) \(\approx\) \(3.252183559\)
\(L(1)\) \(\approx\) \(1.758954199\)
\(L(1)\) \(\approx\) \(1.758954199\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−24.728584582383577172349426370956, −23.97935479783368948737549443518, −22.73319330326781274123768502218, −22.52720483688479565032762206096, −21.567921275321626239073881079985, −20.911986921194334325508717217, −19.64150086595153930481416869231, −18.65349410583989161715618229915, −17.42415523883608877180986362593, −16.65853163195755425405099762109, −16.01239032704497073610235849707, −14.80586488948589857479288664267, −13.82670762718107752244126315915, −12.79247362040256614834077313524, −12.384643237685290123243491399247, −11.19428518540795196261440616560, −10.150253135920420680162002437721, −9.53082186708427799200772267505, −7.33365656747589738852375979319, −6.6797377282846220207047994133, −5.55950917617441393898067824600, −5.173623926636205370380799165553, −3.64511869772090606662235627805, −2.452519624474152661059243817581, −1.01378058330067191667557590446, 1.01378058330067191667557590446, 2.452519624474152661059243817581, 3.64511869772090606662235627805, 5.173623926636205370380799165553, 5.55950917617441393898067824600, 6.6797377282846220207047994133, 7.33365656747589738852375979319, 9.53082186708427799200772267505, 10.150253135920420680162002437721, 11.19428518540795196261440616560, 12.384643237685290123243491399247, 12.79247362040256614834077313524, 13.82670762718107752244126315915, 14.80586488948589857479288664267, 16.01239032704497073610235849707, 16.65853163195755425405099762109, 17.42415523883608877180986362593, 18.65349410583989161715618229915, 19.64150086595153930481416869231, 20.911986921194334325508717217, 21.567921275321626239073881079985, 22.52720483688479565032762206096, 22.73319330326781274123768502218, 23.97935479783368948737549443518, 24.728584582383577172349426370956

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