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Properties

Label 1-389-389.73-r0-0-0
Degree $1$
Conductor $389$
Sign $0.353 + 0.935i$
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  + (0.997 − 0.0647i)2-s + (0.271 + 0.962i)3-s + (0.991 − 0.129i)4-s + (0.665 + 0.746i)5-s + (0.333 + 0.942i)6-s + (−0.735 − 0.677i)7-s + (0.981 − 0.193i)8-s + (−0.852 + 0.523i)9-s + (0.712 + 0.701i)10-s + (0.145 + 0.989i)11-s + (0.393 + 0.919i)12-s + (−0.481 − 0.876i)13-s + (−0.777 − 0.628i)14-s + (−0.536 + 0.843i)15-s + (0.966 − 0.256i)16-s + (0.712 + 0.701i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0647i)2-s + (0.271 + 0.962i)3-s + (0.991 − 0.129i)4-s + (0.665 + 0.746i)5-s + (0.333 + 0.942i)6-s + (−0.735 − 0.677i)7-s + (0.981 − 0.193i)8-s + (−0.852 + 0.523i)9-s + (0.712 + 0.701i)10-s + (0.145 + 0.989i)11-s + (0.393 + 0.919i)12-s + (−0.481 − 0.876i)13-s + (−0.777 − 0.628i)14-s + (−0.536 + 0.843i)15-s + (0.966 − 0.256i)16-s + (0.712 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.256197109 + 1.559123714i\)
\(L(\frac12)\) \(\approx\) \(2.256197109 + 1.559123714i\)
\(L(1)\) \(\approx\) \(1.950722982 + 0.7567197255i\)
\(L(1)\) \(\approx\) \(1.950722982 + 0.7567197255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.997 - 0.0647i)T \)
3 \( 1 + (0.271 + 0.962i)T \)
5 \( 1 + (0.665 + 0.746i)T \)
7 \( 1 + (-0.735 - 0.677i)T \)
11 \( 1 + (0.145 + 0.989i)T \)
13 \( 1 + (-0.481 - 0.876i)T \)
17 \( 1 + (0.712 + 0.701i)T \)
19 \( 1 + (-0.302 + 0.953i)T \)
23 \( 1 + (0.665 - 0.746i)T \)
29 \( 1 + (-0.995 - 0.0970i)T \)
31 \( 1 + (0.925 - 0.378i)T \)
37 \( 1 + (0.452 + 0.891i)T \)
41 \( 1 + (-0.302 - 0.953i)T \)
43 \( 1 + (-0.884 - 0.466i)T \)
47 \( 1 + (-0.302 - 0.953i)T \)
53 \( 1 + (0.208 - 0.977i)T \)
59 \( 1 + (-0.937 - 0.348i)T \)
61 \( 1 + (-0.481 - 0.876i)T \)
67 \( 1 + (-0.986 + 0.161i)T \)
71 \( 1 + (0.991 - 0.129i)T \)
73 \( 1 + (0.966 + 0.256i)T \)
79 \( 1 + (0.925 + 0.378i)T \)
83 \( 1 + (-0.986 - 0.161i)T \)
89 \( 1 + (-0.957 + 0.287i)T \)
97 \( 1 + (0.991 + 0.129i)T \)
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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

−24.34165526464002573642779296493, −23.617522869324505249841929618290, −22.638260767063028184418820047042, −21.56540399550226698216300323312, −21.1571258924292067189697502713, −19.83353414018555446034808875052, −19.337607593827459545624691191915, −18.30917986425365907385659355311, −16.9636135328383867056960390982, −16.45739927082330526862876324266, −15.27689903319174773741387449848, −14.17378855930537381967602585286, −13.52007609982976477665415402100, −12.8664876437521087393084445566, −12.03207479833172614414925531002, −11.2920425526014106067880304397, −9.5439814568972514057719663635, −8.806575713444038615814633552955, −7.526227050816954272681340391328, −6.47243639129479518595875889819, −5.82691353658386783835397469510, −4.847490631912245758521007590803, −3.23490953175931253245820037904, −2.46338101653674062978820560806, −1.25379561465331380440704950852, 2.002232329910730187661498340824, 3.110730424318626235196761798411, 3.79820442488297574560250079241, 4.95095336108434688787372091949, 5.93870194006095621407435286210, 6.88638372694038937556640767130, 7.974821485312959817147504529569, 9.85346892091333153886833436640, 10.10993801251736086750412917122, 10.9112091102503746562348392035, 12.28968226633196153835408841737, 13.18384397707934419020350596174, 14.105673121761318723830806031268, 14.95548981405743386761080997241, 15.30072684755124363756940540145, 16.83156312640038892151197051565, 17.0269108039457314484889336785, 18.77081672208075016715610017762, 19.77343051320985589940650340558, 20.54929078565986780709039265882, 21.18442004509081285309042212013, 22.28846074876745162826449733634, 22.64223362874755646282530139763, 23.29220632501693603184369536934, 24.84571589502388626443062723635

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