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Properties

Label 2-855-15.2-c1-0-25
Degree $2$
Conductor $855$
Sign $0.806 + 0.591i$
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  + (1.20 − 1.20i)2-s − 0.924i·4-s + (1.20 − 1.88i)5-s + (2.18 + 2.18i)7-s + (1.30 + 1.30i)8-s + (−0.812 − 3.73i)10-s + 6.39i·11-s + (3.18 − 3.18i)13-s + 5.28·14-s + 4.99·16-s + (−4.92 + 4.92i)17-s − i·19-s + (−1.73 − 1.11i)20-s + (7.73 + 7.73i)22-s + (−1.62 − 1.62i)23-s + ⋯
L(s)  = 1  + (0.855 − 0.855i)2-s − 0.462i·4-s + (0.540 − 0.841i)5-s + (0.825 + 0.825i)7-s + (0.459 + 0.459i)8-s + (−0.256 − 1.18i)10-s + 1.92i·11-s + (0.883 − 0.883i)13-s + 1.41·14-s + 1.24·16-s + (−1.19 + 1.19i)17-s − 0.229i·19-s + (−0.388 − 0.250i)20-s + (1.64 + 1.64i)22-s + (−0.338 − 0.338i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.806 + 0.591i$
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: \((2,\ 855,\ (\ :1/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85234 - 0.934659i\)
\(L(\frac12)\) \(\approx\) \(2.85234 - 0.934659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.88i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-1.20 + 1.20i)T - 2iT^{2} \)
7 \( 1 + (-2.18 - 2.18i)T + 7iT^{2} \)
11 \( 1 - 6.39iT - 11T^{2} \)
13 \( 1 + (-3.18 + 3.18i)T - 13iT^{2} \)
17 \( 1 + (4.92 - 4.92i)T - 17iT^{2} \)
23 \( 1 + (1.62 + 1.62i)T + 23iT^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 + (6.56 + 6.56i)T + 37iT^{2} \)
41 \( 1 + 5.32iT - 41T^{2} \)
43 \( 1 + (-4.02 + 4.02i)T - 43iT^{2} \)
47 \( 1 + (2.10 - 2.10i)T - 47iT^{2} \)
53 \( 1 + (7.08 + 7.08i)T + 53iT^{2} \)
59 \( 1 + 4.04T + 59T^{2} \)
61 \( 1 + 2.29T + 61T^{2} \)
67 \( 1 + (3.38 + 3.38i)T + 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (-5.68 + 5.68i)T - 73iT^{2} \)
79 \( 1 - 0.156iT - 79T^{2} \)
83 \( 1 + (-8.12 - 8.12i)T + 83iT^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (-0.439 - 0.439i)T + 97iT^{2} \)
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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

−10.45468669835365243717772697222, −9.254316158374347131083899679574, −8.505867066083111839702900982630, −7.73989850335794677437675319779, −6.30093950836340200529005110252, −5.25974632046301523454785548616, −4.71845617138921207802711717562, −3.83440468046013658596891285774, −2.16159398193072171034659577413, −1.81471694074854427549700048966, 1.35886906501131070552840472171, 3.11004102059046477477080273261, 4.09199304396202806450300150437, 5.04185577555209907823464398900, 6.09276265664119798104130591156, 6.54327097945996898311022595874, 7.41541393132116901666031715707, 8.352715556417986367290285355033, 9.351211110930155458419352540592, 10.52098462622647723624578600531

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