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Properties

Label 2-855-15.2-c1-0-11
Degree $2$
Conductor $855$
Sign $0.948 - 0.318i$
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 + (−1.67 + 1.48i)5-s + (0.633 + 0.633i)7-s + (−0.267 + 4.46i)10-s + 5.79i·11-s + (−2.73 + 2.73i)13-s + 1.79·14-s + 3.99·16-s + (−0.328 + 0.328i)17-s − i·19-s + (2.96 + 3.34i)20-s + (8.19 + 8.19i)22-s + (2.44 + 2.44i)23-s + (0.598 − 4.96i)25-s + 7.72i·26-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)2-s − 1.00i·4-s + (−0.748 + 0.663i)5-s + (0.239 + 0.239i)7-s + (−0.0847 + 1.41i)10-s + 1.74i·11-s + (−0.757 + 0.757i)13-s + 0.479·14-s + 0.999·16-s + (−0.0795 + 0.0795i)17-s − 0.229i·19-s + (0.663 + 0.748i)20-s + (1.74 + 1.74i)22-s + (0.510 + 0.510i)23-s + (0.119 − 0.992i)25-s + 1.51i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.948 - 0.318i$
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: \((2,\ 855,\ (\ :1/2),\ 0.948 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15605 + 0.352041i\)
\(L(\frac12)\) \(\approx\) \(2.15605 + 0.352041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 1.48i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-1.41 + 1.41i)T - 2iT^{2} \)
7 \( 1 + (-0.633 - 0.633i)T + 7iT^{2} \)
11 \( 1 - 5.79iT - 11T^{2} \)
13 \( 1 + (2.73 - 2.73i)T - 13iT^{2} \)
17 \( 1 + (0.328 - 0.328i)T - 17iT^{2} \)
23 \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \)
29 \( 1 - 5.93T + 29T^{2} \)
31 \( 1 + 7.46T + 31T^{2} \)
37 \( 1 + (3.46 + 3.46i)T + 37iT^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (-2.09 + 2.09i)T - 43iT^{2} \)
47 \( 1 + (0.189 - 0.189i)T - 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 3.58T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 + (-2.53 - 2.53i)T + 67iT^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (9.09 - 9.09i)T - 73iT^{2} \)
79 \( 1 + 6.53iT - 79T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \)
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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

−10.39656478676997256297409065284, −9.788343022660395273206471206334, −8.590961737472469533108227637626, −7.33735989513378650657798552790, −6.98748227890536452454264183410, −5.40370491460258879166205471642, −4.55015993180686136204153140540, −3.91674185980054559698541460145, −2.70183120521033932881759661116, −1.89472880525799258899108023071, 0.793283810688332276062479793358, 3.12839988486698643636974009704, 3.98068806821498141724367472281, 5.00772111859110362724207109108, 5.54307475510271426198410396395, 6.58514135525643728631014462377, 7.51892361689601571815559171183, 8.192634049129511079587530487394, 8.893037075846314073019086485149, 10.25869290368667600408945943579

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