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Properties

Label 2-855-855.94-c0-0-7
Degree $2$
Conductor $855$
Sign $0.642 + 0.766i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
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  + (0.258 + 0.448i)2-s + (0.258 − 0.965i)3-s + (0.366 − 0.633i)4-s + (0.5 − 0.866i)5-s + (0.5 − 0.133i)6-s + 0.896·8-s + (−0.866 − 0.499i)9-s + 0.517·10-s + (0.866 + 1.5i)11-s + (−0.517 − 0.517i)12-s + (−0.965 + 1.67i)13-s + (−0.707 − 0.707i)15-s + (−0.133 − 0.232i)16-s − 0.517i·18-s − 19-s + (−0.366 − 0.633i)20-s + ⋯
L(s)  = 1  + (0.258 + 0.448i)2-s + (0.258 − 0.965i)3-s + (0.366 − 0.633i)4-s + (0.5 − 0.866i)5-s + (0.5 − 0.133i)6-s + 0.896·8-s + (−0.866 − 0.499i)9-s + 0.517·10-s + (0.866 + 1.5i)11-s + (−0.517 − 0.517i)12-s + (−0.965 + 1.67i)13-s + (−0.707 − 0.707i)15-s + (−0.133 − 0.232i)16-s − 0.517i·18-s − 19-s + (−0.366 − 0.633i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412866339\)
\(L(\frac12)\) \(\approx\) \(1.412866339\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
good2 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.93T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.517T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.986809152936317890777492205229, −9.379682683670594814945563458857, −8.613529823221291036122543074067, −7.33022350663419562065227978604, −6.83784677248083785496667848853, −6.15809688135501068975143592321, −4.97854456135779370813427102000, −4.29260177986610840004307763572, −2.04724174466862553820660629965, −1.71910913913388906295208041814, 2.33037960527582258482904393235, 3.21896062138567563141960866273, 3.73209229114819447036188707795, 5.11957012827856894743233852023, 6.03759181940961949871871405149, 7.08001283122865542553024218197, 8.153971568924370204163226123099, 8.786261696236998893886144130901, 10.01765495444971854586541390017, 10.52875446756698915105391544648

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