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Properties

Label 2-855-15.8-c1-0-22
Degree $2$
Conductor $855$
Sign $-0.990 - 0.139i$
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  + (−1.41 − 1.41i)2-s + 2.00i·4-s + (0.448 − 2.19i)5-s + (2.36 − 2.36i)7-s + (−3.73 + 2.46i)10-s − 1.55i·11-s + (0.732 + 0.732i)13-s − 6.69·14-s + 3.99·16-s + (−4.57 − 4.57i)17-s + i·19-s + (4.38 + 0.896i)20-s + (−2.19 + 2.19i)22-s + (2.44 − 2.44i)23-s + (−4.59 − 1.96i)25-s − 2.07i·26-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)2-s + 1.00i·4-s + (0.200 − 0.979i)5-s + (0.894 − 0.894i)7-s + (−1.18 + 0.779i)10-s − 0.468i·11-s + (0.203 + 0.203i)13-s − 1.78·14-s + 0.999·16-s + (−1.10 − 1.10i)17-s + 0.229i·19-s + (0.979 + 0.200i)20-s + (−0.468 + 0.468i)22-s + (0.510 − 0.510i)23-s + (−0.919 − 0.392i)25-s − 0.406i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0603121 + 0.860547i\)
\(L(\frac12)\) \(\approx\) \(0.0603121 + 0.860547i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.448 + 2.19i)T \)
19 \( 1 - iT \)
good2 \( 1 + (1.41 + 1.41i)T + 2iT^{2} \)
7 \( 1 + (-2.36 + 2.36i)T - 7iT^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 + (-0.732 - 0.732i)T + 13iT^{2} \)
17 \( 1 + (4.57 + 4.57i)T + 17iT^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 - 8.76T + 29T^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (3.09 + 3.09i)T + 43iT^{2} \)
47 \( 1 + (-2.63 - 2.63i)T + 47iT^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 + (-9.46 + 9.46i)T - 67iT^{2} \)
71 \( 1 - 0.757iT - 71T^{2} \)
73 \( 1 + (3.90 + 3.90i)T + 73iT^{2} \)
79 \( 1 - 13.4iT - 79T^{2} \)
83 \( 1 + (-0.757 + 0.757i)T - 83iT^{2} \)
89 \( 1 + 8.76T + 89T^{2} \)
97 \( 1 + (5.66 - 5.66i)T - 97iT^{2} \)
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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

−9.670602408605097508814834804380, −9.016565546564050166312399566787, −8.353070905662392134554706592801, −7.61519269112524189424366122894, −6.32958960787657719250993097572, −5.00607144157119136609480061433, −4.29005231560899561709172934701, −2.79483570346346873749141457088, −1.54630250482827433975175338122, −0.61695294130644760877765392275, 1.75712474657630524395957549895, 3.04069773862045940234842816532, 4.59969046645853080348702543714, 5.80896160556610727981514315156, 6.46860100926071715488518746017, 7.25505024324068241861885216674, 8.171088489835134669853109195254, 8.692329712559667251491492095619, 9.586973198648706763450935511138, 10.43723070647169722056854556080

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