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Properties

Label 1-389-389.67-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.471 + 0.881i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (−0.735 + 0.677i)2-s + (−0.689 + 0.724i)3-s + (0.0808 − 0.996i)4-s + (0.966 + 0.256i)5-s + (0.0161 − 0.999i)6-s + (0.756 + 0.653i)7-s + (0.616 + 0.787i)8-s + (−0.0485 − 0.998i)9-s + (−0.884 + 0.466i)10-s + (−0.777 − 0.628i)11-s + (0.665 + 0.746i)12-s + (0.271 + 0.962i)13-s + (−0.999 + 0.0323i)14-s + (−0.852 + 0.523i)15-s + (−0.986 − 0.161i)16-s + (−0.884 + 0.466i)17-s + ⋯
L(s)  = 1  + (−0.735 + 0.677i)2-s + (−0.689 + 0.724i)3-s + (0.0808 − 0.996i)4-s + (0.966 + 0.256i)5-s + (0.0161 − 0.999i)6-s + (0.756 + 0.653i)7-s + (0.616 + 0.787i)8-s + (−0.0485 − 0.998i)9-s + (−0.884 + 0.466i)10-s + (−0.777 − 0.628i)11-s + (0.665 + 0.746i)12-s + (0.271 + 0.962i)13-s + (−0.999 + 0.0323i)14-s + (−0.852 + 0.523i)15-s + (−0.986 − 0.161i)16-s + (−0.884 + 0.466i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.471 + 0.881i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4523349629 + 0.7547396074i\)
\(L(\frac12)\) \(\approx\) \(0.4523349629 + 0.7547396074i\)
\(L(1)\) \(\approx\) \(0.6255162581 + 0.4504033257i\)
\(L(1)\) \(\approx\) \(0.6255162581 + 0.4504033257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.735 + 0.677i)T \)
3 \( 1 + (-0.689 + 0.724i)T \)
5 \( 1 + (0.966 + 0.256i)T \)
7 \( 1 + (0.756 + 0.653i)T \)
11 \( 1 + (-0.777 - 0.628i)T \)
13 \( 1 + (0.271 + 0.962i)T \)
17 \( 1 + (-0.884 + 0.466i)T \)
19 \( 1 + (0.925 - 0.378i)T \)
23 \( 1 + (0.966 - 0.256i)T \)
29 \( 1 + (0.898 - 0.438i)T \)
31 \( 1 + (-0.240 + 0.970i)T \)
37 \( 1 + (-0.995 - 0.0970i)T \)
41 \( 1 + (0.925 + 0.378i)T \)
43 \( 1 + (-0.641 - 0.767i)T \)
47 \( 1 + (0.925 + 0.378i)T \)
53 \( 1 + (0.997 + 0.0647i)T \)
59 \( 1 + (-0.816 + 0.577i)T \)
61 \( 1 + (0.271 + 0.962i)T \)
67 \( 1 + (-0.957 + 0.287i)T \)
71 \( 1 + (0.0808 - 0.996i)T \)
73 \( 1 + (-0.986 + 0.161i)T \)
79 \( 1 + (-0.240 - 0.970i)T \)
83 \( 1 + (-0.957 - 0.287i)T \)
89 \( 1 + (0.208 + 0.977i)T \)
97 \( 1 + (0.0808 + 0.996i)T \)
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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

−24.45014459550950316879417119753, −23.160282051247217297198656400507, −22.444239783415612171971310128726, −21.38651842941066283833808168190, −20.53992011516499661391594856888, −19.996483902851929413632295619949, −18.58687925194526313699995893031, −17.92225323953310228574355825834, −17.55041709945700467135150987316, −16.74214455303682513235425812687, −15.648674151025184268731124512454, −13.92980431562593517049068057508, −13.21752279594674325384400331465, −12.55586496103672825784712479143, −11.407734202956723733539392339689, −10.63354394835359802612311850910, −9.9553136146223557137257078379, −8.65419440050373228566673207127, −7.65399014555914548384420327352, −6.94867508860906092488549476367, −5.49648130364056884262771036580, −4.63910707647947716047020324577, −2.803596908447755244189618793248, −1.77210855919250699994511500476, −0.837152349310015581442121574590, 1.32267744171561156159258946727, 2.664828859397393119589624728163, 4.648525724365003745938471323678, 5.4128882586014386519978627941, 6.1633734137170899440283728691, 7.122266981460539025666892082955, 8.77352287999229605970559744259, 9.05569849375343476773926942466, 10.374098810345177984212081916418, 10.87876640482384426871168500694, 11.82695367264357791241546436688, 13.48334695718168521582082851750, 14.34638027683931585090608056703, 15.289214555984247483851347969128, 16.003078217374361274587860636764, 16.90529544854299724906640786707, 17.79304690895670329793556981956, 18.17403621647032059142570598060, 19.17130151847540832533628575508, 20.65500955894412958506164459034, 21.35219635614036240500810625936, 22.0344559799200831043788514726, 23.16101034049457326754262141045, 24.08471907079738052787408648051, 24.70531161930360244113639193842

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