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Properties

Label 2-855-15.2-c1-0-16
Degree $2$
Conductor $855$
Sign $-0.0632 - 0.997i$
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  + (−1.95 + 1.95i)2-s − 5.67i·4-s + (2.21 − 0.279i)5-s + (0.856 + 0.856i)7-s + (7.20 + 7.20i)8-s + (−3.79 + 4.89i)10-s + 5.46i·11-s + (2.07 − 2.07i)13-s − 3.35·14-s − 16.8·16-s + (4.51 − 4.51i)17-s + i·19-s + (−1.58 − 12.5i)20-s + (−10.6 − 10.6i)22-s + (−1.88 − 1.88i)23-s + ⋯
L(s)  = 1  + (−1.38 + 1.38i)2-s − 2.83i·4-s + (0.992 − 0.124i)5-s + (0.323 + 0.323i)7-s + (2.54 + 2.54i)8-s + (−1.20 + 1.54i)10-s + 1.64i·11-s + (0.574 − 0.574i)13-s − 0.896·14-s − 4.22·16-s + (1.09 − 1.09i)17-s + 0.229i·19-s + (−0.354 − 2.81i)20-s + (−2.28 − 2.28i)22-s + (−0.393 − 0.393i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0632 - 0.997i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.0632 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685104 + 0.729916i\)
\(L(\frac12)\) \(\approx\) \(0.685104 + 0.729916i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.279i)T \)
19 \( 1 - iT \)
good2 \( 1 + (1.95 - 1.95i)T - 2iT^{2} \)
7 \( 1 + (-0.856 - 0.856i)T + 7iT^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + (-2.07 + 2.07i)T - 13iT^{2} \)
17 \( 1 + (-4.51 + 4.51i)T - 17iT^{2} \)
23 \( 1 + (1.88 + 1.88i)T + 23iT^{2} \)
29 \( 1 + 0.00315T + 29T^{2} \)
31 \( 1 + 0.931T + 31T^{2} \)
37 \( 1 + (0.220 + 0.220i)T + 37iT^{2} \)
41 \( 1 - 8.50iT - 41T^{2} \)
43 \( 1 + (-1.78 + 1.78i)T - 43iT^{2} \)
47 \( 1 + (0.566 - 0.566i)T - 47iT^{2} \)
53 \( 1 + (-6.18 - 6.18i)T + 53iT^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + (-9.71 - 9.71i)T + 67iT^{2} \)
71 \( 1 - 3.58iT - 71T^{2} \)
73 \( 1 + (-5.50 + 5.50i)T - 73iT^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (1.80 + 1.80i)T + 83iT^{2} \)
89 \( 1 + 3.93T + 89T^{2} \)
97 \( 1 + (-5.28 - 5.28i)T + 97iT^{2} \)
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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

−9.957453672891377066572482343663, −9.537036583878400313915436562389, −8.687973178800892350055029532040, −7.82558039300961463037707667324, −7.13531704047780561738970321400, −6.23506476786177191171987196904, −5.43481559306309288777402429957, −4.77755031447644225323443529207, −2.27492124551591776460804992403, −1.17980251240423029136413903201, 0.986606180292959537378550833786, 1.92547384242925768797707021746, 3.17768219305565393268301991576, 3.93320946687872409736127630050, 5.65382082331793586851168123192, 6.73624206094172370828753925200, 7.931630999592316008226688768529, 8.519640302010572903324335956385, 9.265956774243256563563283512393, 10.01316818622108235991092841822

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