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Properties

Label 2-855-95.4-c1-0-26
Degree $2$
Conductor $855$
Sign $0.998 + 0.0625i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0854 − 0.234i)2-s + (1.48 + 1.24i)4-s + (2.12 − 0.708i)5-s + (−3.42 − 1.98i)7-s + (0.851 − 0.491i)8-s + (0.0148 − 0.558i)10-s + (1.56 + 2.71i)11-s + (2.61 + 0.461i)13-s + (−0.757 + 0.635i)14-s + (0.630 + 3.57i)16-s + (−0.771 + 2.12i)17-s + (3.76 − 2.19i)19-s + (4.03 + 1.58i)20-s + (0.770 − 0.135i)22-s + (4.87 − 5.81i)23-s + ⋯
L(s)  = 1  + (0.0604 − 0.165i)2-s + (0.742 + 0.622i)4-s + (0.948 − 0.316i)5-s + (−1.29 − 0.748i)7-s + (0.301 − 0.173i)8-s + (0.00471 − 0.176i)10-s + (0.472 + 0.818i)11-s + (0.726 + 0.128i)13-s + (−0.202 + 0.169i)14-s + (0.157 + 0.893i)16-s + (−0.187 + 0.514i)17-s + (0.864 − 0.502i)19-s + (0.901 + 0.355i)20-s + (0.164 − 0.0289i)22-s + (1.01 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.998 + 0.0625i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.998 + 0.0625i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15683 - 0.0675578i\)
\(L(\frac12)\) \(\approx\) \(2.15683 - 0.0675578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.12 + 0.708i)T \)
19 \( 1 + (-3.76 + 2.19i)T \)
good2 \( 1 + (-0.0854 + 0.234i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (3.42 + 1.98i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.56 - 2.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.61 - 0.461i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.771 - 2.12i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.87 + 5.81i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.28 - 0.466i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.447 - 0.774i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.62iT - 37T^{2} \)
41 \( 1 + (1.08 + 6.14i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.13 - 1.35i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.0603 - 0.165i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.35 - 5.19i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-7.34 - 2.67i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.796 + 0.668i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.20 + 14.2i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.99 - 6.70i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-4.11 + 0.726i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.94 + 11.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (11.4 + 6.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.257 + 1.46i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.26 - 14.4i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.26235919199988571769845825741, −9.382242818570579444839430708342, −8.662069785633598202550880821544, −7.33851525984918936951348557150, −6.68803565700158343764086268796, −6.13552695030481967774150668972, −4.67822020775873516436505782877, −3.60602822252348872513763053642, −2.66740690119685371423623317536, −1.33218315859597434489025288612, 1.31530299492264211639519317432, 2.69509789898255668916228418486, 3.42799858104915743731960926009, 5.45221916065187063887961271176, 5.80049567711747763770037305147, 6.56321463616531279561416179113, 7.29188538367608986596618592223, 8.756486482525800584420105725698, 9.531741079727077669921468718131, 9.971655107538762964661746697809

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