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Properties

Label 2-855-95.54-c1-0-9
Degree $2$
Conductor $855$
Sign $-0.999 - 0.0184i$
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  + (−0.288 + 0.344i)2-s + (0.312 + 1.77i)4-s + (0.869 + 2.05i)5-s + (−1.78 + 1.03i)7-s + (−1.47 − 0.853i)8-s + (−0.960 − 0.295i)10-s + (−1.15 + 1.99i)11-s + (1.50 − 4.13i)13-s + (0.160 − 0.911i)14-s + (−2.65 + 0.967i)16-s + (−3.51 + 4.19i)17-s + (−3.07 + 3.09i)19-s + (−3.37 + 2.18i)20-s + (−0.354 − 0.973i)22-s + (5.72 − 1.01i)23-s + ⋯
L(s)  = 1  + (−0.204 + 0.243i)2-s + (0.156 + 0.885i)4-s + (0.389 + 0.921i)5-s + (−0.674 + 0.389i)7-s + (−0.522 − 0.301i)8-s + (−0.303 − 0.0934i)10-s + (−0.347 + 0.602i)11-s + (0.417 − 1.14i)13-s + (0.0429 − 0.243i)14-s + (−0.664 + 0.241i)16-s + (−0.853 + 1.01i)17-s + (−0.704 + 0.709i)19-s + (−0.754 + 0.488i)20-s + (−0.0755 − 0.207i)22-s + (1.19 − 0.210i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.999 - 0.0184i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.999 - 0.0184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00853916 + 0.925783i\)
\(L(\frac12)\) \(\approx\) \(0.00853916 + 0.925783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.869 - 2.05i)T \)
19 \( 1 + (3.07 - 3.09i)T \)
good2 \( 1 + (0.288 - 0.344i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (1.78 - 1.03i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.15 - 1.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.50 + 4.13i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.51 - 4.19i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.72 + 1.01i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.21 + 3.53i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.378 + 0.656i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.22iT - 37T^{2} \)
41 \( 1 + (6.14 - 2.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.11 + 0.549i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.08 + 4.87i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-3.79 + 0.668i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-1.95 - 1.63i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.72 - 9.78i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.09 - 1.30i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.71 - 9.70i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.49 - 9.60i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-2.26 + 0.824i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (6.11 - 3.53i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.19 - 0.798i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.38 - 5.22i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.49098968616665355372896688516, −9.849093750400723498283939476798, −8.717086436095873919651929365182, −8.125375223237412500407041070038, −7.05962241432934562302795788234, −6.47817470623305737335980253599, −5.62446154415085534659936030040, −4.06253172439596287400519175404, −3.10517786323324195312435682886, −2.27932433435950468844649053509, 0.45605878318694118063468912433, 1.73527513366957080676114509431, 3.02753795156432692385211078063, 4.60357912363092937591959191685, 5.18494724919403117421584122535, 6.48225023944764386700677146094, 6.78332348966208700157045732682, 8.461925741182679454372475195665, 9.095621921843877020175553684020, 9.616679815084131508102963595170

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