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Properties

Label 2-855-95.74-c1-0-36
Degree $2$
Conductor $855$
Sign $0.947 + 0.319i$
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  + (1.93 + 0.340i)2-s + (1.73 + 0.632i)4-s + (1.25 − 1.85i)5-s + (−0.586 + 0.338i)7-s + (−0.257 − 0.148i)8-s + (3.05 − 3.15i)10-s + (1.42 − 2.46i)11-s + (3.06 + 3.65i)13-s + (−1.24 + 0.454i)14-s + (−3.27 − 2.75i)16-s + (5.10 + 0.900i)17-s + (3.00 − 3.15i)19-s + (3.34 − 2.42i)20-s + (3.58 − 4.26i)22-s + (−0.359 + 0.987i)23-s + ⋯
L(s)  = 1  + (1.36 + 0.240i)2-s + (0.868 + 0.316i)4-s + (0.560 − 0.828i)5-s + (−0.221 + 0.127i)7-s + (−0.0909 − 0.0525i)8-s + (0.965 − 0.996i)10-s + (0.428 − 0.741i)11-s + (0.849 + 1.01i)13-s + (−0.333 + 0.121i)14-s + (−0.819 − 0.687i)16-s + (1.23 + 0.218i)17-s + (0.690 − 0.723i)19-s + (0.748 − 0.542i)20-s + (0.763 − 0.910i)22-s + (−0.0749 + 0.205i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.39682 - 0.557411i\)
\(L(\frac12)\) \(\approx\) \(3.39682 - 0.557411i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 1.85i)T \)
19 \( 1 + (-3.00 + 3.15i)T \)
good2 \( 1 + (-1.93 - 0.340i)T + (1.87 + 0.684i)T^{2} \)
7 \( 1 + (0.586 - 0.338i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.42 + 2.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.06 - 3.65i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-5.10 - 0.900i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.359 - 0.987i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.247 + 1.40i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.135 + 0.234i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.603iT - 37T^{2} \)
41 \( 1 + (5.15 + 4.32i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.92 - 5.28i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.77 - 1.37i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.35 - 6.47i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.75 - 9.96i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.02 - 2.55i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-4.05 + 0.714i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.14 - 2.60i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (10.3 - 12.3i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (11.3 + 9.54i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.2 - 7.09i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.31 + 5.29i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-5.64 - 0.994i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.03552730972695335996343439292, −9.208009463226448923930699642028, −8.594488621947369093780347878179, −7.28112570927295582602423097676, −6.15566376719943571281240276365, −5.79966902855978169803811723375, −4.80826239165845764004014530420, −3.92598133570811703677575018887, −2.96628309321762873184693704118, −1.29976568848759955320377744613, 1.76456662869032415160626727193, 3.21854614088579426688605966629, 3.52454243637506136210861372944, 4.95440535005535074283839670977, 5.72954640597343036174579935487, 6.43604035212049959522746120178, 7.37749891219508759873729797004, 8.469842524199548818009465816752, 9.797654388775636232324068098062, 10.21463224451059600269558061655

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