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Properties

Label 2-855-15.2-c1-0-14
Degree $2$
Conductor $855$
Sign $0.996 + 0.0799i$
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  + (−0.707 + 0.707i)2-s + 0.999i·4-s + (−2.12 + 0.707i)5-s + (−3 − 3i)7-s + (−2.12 − 2.12i)8-s + (0.999 − 2i)10-s + 4.24i·11-s + (2 − 2i)13-s + 4.24·14-s + 1.00·16-s − i·19-s + (−0.707 − 2.12i)20-s + (−3 − 3i)22-s + (3.99 − 3i)25-s + 2.82i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + 0.499i·4-s + (−0.948 + 0.316i)5-s + (−1.13 − 1.13i)7-s + (−0.750 − 0.750i)8-s + (0.316 − 0.632i)10-s + 1.27i·11-s + (0.554 − 0.554i)13-s + 1.13·14-s + 0.250·16-s − 0.229i·19-s + (−0.158 − 0.474i)20-s + (−0.639 − 0.639i)22-s + (0.799 − 0.600i)25-s + 0.554i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.996 + 0.0799i$
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: \((2,\ 855,\ (\ :1/2),\ 0.996 + 0.0799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633934 - 0.0253682i\)
\(L(\frac12)\) \(\approx\) \(0.633934 - 0.0253682i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.707i)T \)
19 \( 1 + iT \)
good2 \( 1 + (0.707 - 0.707i)T - 2iT^{2} \)
7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 47iT^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + (-6 - 6i)T + 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + (-7.07 - 7.07i)T + 83iT^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−10.10273312798393362442259811934, −9.293452225021782161514515047683, −8.251951362349695470088946759804, −7.53774315863346037094458741362, −6.94770768299676503265394182508, −6.32442912438706596323775097407, −4.52873824571890574084494965097, −3.74973781943591857431402333583, −2.92038691376442548520436595860, −0.49046628615534438045276523936, 0.947935815525567468503950032762, 2.65594357544365231294812638604, 3.48136554584520222061164588376, 4.87556956241656723113418404977, 6.04137234357231176976273252538, 6.42939782061921232132135881240, 8.054422392783333211062545654576, 8.711283118240967325251898640562, 9.258260858313078094690758086201, 10.12370387221559014253927942939

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