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Properties

Label 2-855-5.4-c1-0-18
Degree $2$
Conductor $855$
Sign $0.868 - 0.495i$
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  + 0.184i·2-s + 1.96·4-s + (−1.94 + 1.10i)5-s − 1.90i·7-s + 0.733i·8-s + (−0.204 − 0.359i)10-s + 3.94·11-s + 4.53i·13-s + 0.353·14-s + 3.79·16-s − 4.68i·17-s − 19-s + (−3.81 + 2.17i)20-s + 0.728i·22-s + 6.18i·23-s + ⋯
L(s)  = 1  + 0.130i·2-s + 0.982·4-s + (−0.868 + 0.495i)5-s − 0.721i·7-s + 0.259i·8-s + (−0.0648 − 0.113i)10-s + 1.18·11-s + 1.25i·13-s + 0.0943·14-s + 0.949·16-s − 1.13i·17-s − 0.229·19-s + (−0.853 + 0.487i)20-s + 0.155i·22-s + 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.868 - 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77055 + 0.469727i\)
\(L(\frac12)\) \(\approx\) \(1.77055 + 0.469727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 - 1.10i)T \)
19 \( 1 + T \)
good2 \( 1 - 0.184iT - 2T^{2} \)
7 \( 1 + 1.90iT - 7T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
13 \( 1 - 4.53iT - 13T^{2} \)
17 \( 1 + 4.68iT - 17T^{2} \)
23 \( 1 - 6.18iT - 23T^{2} \)
29 \( 1 - 5.98T + 29T^{2} \)
31 \( 1 - 7.31T + 31T^{2} \)
37 \( 1 - 8.07iT - 37T^{2} \)
41 \( 1 + 0.0567T + 41T^{2} \)
43 \( 1 - 0.822iT - 43T^{2} \)
47 \( 1 + 7.50iT - 47T^{2} \)
53 \( 1 + 1.28iT - 53T^{2} \)
59 \( 1 + 8.28T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 10.0iT - 73T^{2} \)
79 \( 1 + 4.12T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 - 7.17T + 89T^{2} \)
97 \( 1 + 2.70iT - 97T^{2} \)
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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

−10.35316202541200585452498877549, −9.492791490748728558955267060935, −8.390317052994825492534482641079, −7.44358814114888210727668653605, −6.83145968393831149439977782324, −6.35460949853575775817824692000, −4.72881976773312276692244248092, −3.82253096127920176163854720656, −2.82698299667760794009024950474, −1.31663649179184423456979966240, 1.08376812764315747731126865350, 2.55244067529809864458742099683, 3.59828771499426507132742413456, 4.64654531665847700553453509083, 5.96055118793651989326540255835, 6.51529521449887555949734546723, 7.71426652491748303856067120065, 8.336316814673228084211851426111, 9.119493440730426085580899474396, 10.38359542855089309400401415239

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