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Properties

Label 1-389-389.35-r0-0-0
Degree $1$
Conductor $389$
Sign $0.564 + 0.825i$
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  + (0.925 + 0.378i)2-s + (0.0808 − 0.996i)3-s + (0.712 + 0.701i)4-s + (0.333 + 0.942i)5-s + (0.452 − 0.891i)6-s + (−0.240 + 0.970i)7-s + (0.393 + 0.919i)8-s + (−0.986 − 0.161i)9-s + (−0.0485 + 0.998i)10-s + (−0.641 − 0.767i)11-s + (0.756 − 0.653i)12-s + (0.991 − 0.129i)13-s + (−0.590 + 0.807i)14-s + (0.966 − 0.256i)15-s + (0.0161 + 0.999i)16-s + (−0.0485 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.925 + 0.378i)2-s + (0.0808 − 0.996i)3-s + (0.712 + 0.701i)4-s + (0.333 + 0.942i)5-s + (0.452 − 0.891i)6-s + (−0.240 + 0.970i)7-s + (0.393 + 0.919i)8-s + (−0.986 − 0.161i)9-s + (−0.0485 + 0.998i)10-s + (−0.641 − 0.767i)11-s + (0.756 − 0.653i)12-s + (0.991 − 0.129i)13-s + (−0.590 + 0.807i)14-s + (0.966 − 0.256i)15-s + (0.0161 + 0.999i)16-s + (−0.0485 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.064787950 + 1.089478810i\)
\(L(\frac12)\) \(\approx\) \(2.064787950 + 1.089478810i\)
\(L(1)\) \(\approx\) \(1.766480776 + 0.4692391267i\)
\(L(1)\) \(\approx\) \(1.766480776 + 0.4692391267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.925 + 0.378i)T \)
3 \( 1 + (0.0808 - 0.996i)T \)
5 \( 1 + (0.333 + 0.942i)T \)
7 \( 1 + (-0.240 + 0.970i)T \)
11 \( 1 + (-0.641 - 0.767i)T \)
13 \( 1 + (0.991 - 0.129i)T \)
17 \( 1 + (-0.0485 + 0.998i)T \)
19 \( 1 + (0.271 + 0.962i)T \)
23 \( 1 + (0.333 - 0.942i)T \)
29 \( 1 + (0.834 - 0.550i)T \)
31 \( 1 + (-0.689 + 0.724i)T \)
37 \( 1 + (0.948 - 0.318i)T \)
41 \( 1 + (0.271 - 0.962i)T \)
43 \( 1 + (-0.974 - 0.224i)T \)
47 \( 1 + (0.271 - 0.962i)T \)
53 \( 1 + (-0.302 + 0.953i)T \)
59 \( 1 + (-0.536 - 0.843i)T \)
61 \( 1 + (0.991 - 0.129i)T \)
67 \( 1 + (0.563 + 0.825i)T \)
71 \( 1 + (0.712 + 0.701i)T \)
73 \( 1 + (0.0161 - 0.999i)T \)
79 \( 1 + (-0.689 - 0.724i)T \)
83 \( 1 + (0.563 - 0.825i)T \)
89 \( 1 + (-0.177 + 0.984i)T \)
97 \( 1 + (0.712 - 0.701i)T \)
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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

−23.97603799962195859963003128623, −23.34266144410481354008437632441, −22.6261448579657760173877265001, −21.51421017669432405625524031106, −20.942816597992353293819045395335, −20.12628255240917315659580569588, −19.87708780262050871849537749255, −18.1380591595522569875866988835, −16.98806201285467796559968120503, −16.06317155526831145651983840511, −15.65735640968676708276047479522, −14.43472303086309982853944271131, −13.457425590974741839040750900500, −13.07084612391069275999852719376, −11.6410674028662564992004466304, −10.9128252496973607618694999733, −9.85305650262563767685952016439, −9.3097105369528634207194399133, −7.785814101703265429864044718033, −6.49029763887359419002160868720, −5.218915588470217132762333120619, −4.695988481981462375990417985150, −3.75743391526571138907363767192, −2.64787070888278377404103999118, −1.07987973682010895235718251760, 1.87182534869505673673638019009, 2.80508011497950209828887488768, 3.58764509353213696329186928421, 5.558543707634078294759143135016, 6.03703532527459359270982809582, 6.77423862101357977792729513450, 8.01683135391999489957494512685, 8.63877795636028656514665267994, 10.51712916656960725613379620675, 11.33097241285234093749762298323, 12.35160261455149512246968371114, 13.076746851737627397630360363787, 13.921004632785110857219349519232, 14.65379034782656603087215210070, 15.54703922483507318162904096591, 16.535071460239593170794713764376, 17.731631249194684372626075373962, 18.541282578693544398558612791224, 19.1050770309668123684253411613, 20.41890752948159011551520921127, 21.44204085056181563263933312015, 22.0614339691116124858664573175, 23.12864987008237240357419048578, 23.51262623629186778621763471796, 24.75300850333200737259465826252

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