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Properties

Label 1-399-399.257-r1-0-0
Degree $1$
Conductor $399$
Sign $-0.243 + 0.970i$
Analytic cond. $42.8785$
Root an. cond. $42.8785$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + 20-s + (−0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)26-s + (−0.939 − 0.342i)29-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + 20-s + (−0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)26-s + (−0.939 − 0.342i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.243 + 0.970i$
Analytic conductor: \(42.8785\)
Root analytic conductor: \(42.8785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 399,\ (1:\ ),\ -0.243 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897083969 + 2.431003502i\)
\(L(\frac12)\) \(\approx\) \(1.897083969 + 2.431003502i\)
\(L(1)\) \(\approx\) \(1.523771768 + 0.7623433813i\)
\(L(1)\) \(\approx\) \(1.523771768 + 0.7623433813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−23.71527180853680704806838223344, −22.79409698250779078948194970715, −22.23508842321731306760680555771, −21.44983618941702005693111444274, −20.62921379006470391141749360406, −19.47165463060949527895060469391, −18.98583736256996458368846652097, −18.03767769590822718894516314974, −16.94220448629933485968710200602, −15.59192976069930230131732887, −14.88814434067470300438822327657, −14.12555630955056254833764701957, −13.25519957898194290095700625257, −12.363121381031987326708291395852, −11.15569471632545097221834810923, −10.73422059743056903853802659609, −9.77204238578841687598981185910, −8.52095425432212813081457025606, −7.10395644031210534093585764263, −6.12703117486389830469345511472, −5.409622930963156037611426732542, −3.85453822580741799813794570058, −3.2076485881996041504086179373, −2.091129705390256824631621715035, −0.6768337846858851872849963486, 1.32914830768348015437254836934, 2.716785432608466913516503172344, 4.2264162052675079701789187806, 4.73468314071697428025628908411, 5.851352382107150107867272812834, 6.88483546464292399774312392835, 7.79416530889055543237399059160, 8.978089689672732559683488878657, 9.603353553407017351742405004064, 11.43581287289433006653224616369, 12.03593861800740249235364535662, 13.023521340957214902288773787567, 13.72195041943808274681363434444, 14.68404094139928320795900677374, 15.61020650307502643702110224816, 16.523811810266133349789037455903, 17.080392996730397112039702590437, 18.00078769221188442121111692504, 19.32093503824038086305914149812, 20.51888837618255052212519163939, 20.90992369271579633988758409935, 21.91078113727774383475224398637, 22.86732739311919317032264888149, 23.55800282062283644044265609924, 24.483323484206518221554707220320

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