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Properties

Label 2-855-15.8-c1-0-2
Degree $2$
Conductor $855$
Sign $-0.426 - 0.904i$
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  + (0.0868 + 0.0868i)2-s − 1.98i·4-s + (0.0868 + 2.23i)5-s + (−1.64 + 1.64i)7-s + (0.345 − 0.345i)8-s + (−0.186 + 0.201i)10-s + 2.91i·11-s + (−0.641 − 0.641i)13-s − 0.284·14-s − 3.90·16-s + (−0.606 − 0.606i)17-s + i·19-s + (4.43 − 0.172i)20-s + (−0.253 + 0.253i)22-s + (−5.84 + 5.84i)23-s + ⋯
L(s)  = 1  + (0.0613 + 0.0613i)2-s − 0.992i·4-s + (0.0388 + 0.999i)5-s + (−0.620 + 0.620i)7-s + (0.122 − 0.122i)8-s + (−0.0589 + 0.0637i)10-s + 0.879i·11-s + (−0.177 − 0.177i)13-s − 0.0761·14-s − 0.977·16-s + (−0.147 − 0.147i)17-s + 0.229i·19-s + (0.991 − 0.0385i)20-s + (−0.0540 + 0.0540i)22-s + (−1.21 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.426 - 0.904i$
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: \((2,\ 855,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486025 + 0.766493i\)
\(L(\frac12)\) \(\approx\) \(0.486025 + 0.766493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0868 - 2.23i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-0.0868 - 0.0868i)T + 2iT^{2} \)
7 \( 1 + (1.64 - 1.64i)T - 7iT^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 + (0.641 + 0.641i)T + 13iT^{2} \)
17 \( 1 + (0.606 + 0.606i)T + 17iT^{2} \)
23 \( 1 + (5.84 - 5.84i)T - 23iT^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 + (3.64 - 3.64i)T - 37iT^{2} \)
41 \( 1 - 1.20iT - 41T^{2} \)
43 \( 1 + (-5.25 - 5.25i)T + 43iT^{2} \)
47 \( 1 + (-2.86 - 2.86i)T + 47iT^{2} \)
53 \( 1 + (1.93 - 1.93i)T - 53iT^{2} \)
59 \( 1 - 9.68T + 59T^{2} \)
61 \( 1 - 8.26T + 61T^{2} \)
67 \( 1 + (4.28 - 4.28i)T - 67iT^{2} \)
71 \( 1 + 1.42iT - 71T^{2} \)
73 \( 1 + (1.21 + 1.21i)T + 73iT^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 + (7.59 - 7.59i)T - 83iT^{2} \)
89 \( 1 + 4.51T + 89T^{2} \)
97 \( 1 + (-3.01 + 3.01i)T - 97iT^{2} \)
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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

−10.19368898514610694033665908700, −9.853611800488525493285576248459, −9.035262082833899324435214694128, −7.69047433879113392558742163834, −6.87812231090032816385263995031, −6.09827335767449462776879074879, −5.39691140651677141405753597285, −4.15042993970094346982170820717, −2.86041097822178502764905583549, −1.82376763220811711906147276957, 0.40973512003572233247124247204, 2.30155897063469693884981834608, 3.65152952153777324772322703292, 4.24430648805279224376234479748, 5.42438797638458599005568539033, 6.54429326942448877011872838481, 7.40214050298450641177728691638, 8.461219171297328808631505686476, 8.762736806276143841234608333915, 9.865197290936401708021379012309

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