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Properties

Label 2-855-19.7-c1-0-26
Degree $2$
Conductor $855$
Sign $0.910 + 0.412i$
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  + (0.610 + 1.05i)2-s + (0.253 − 0.439i)4-s + (−0.5 − 0.866i)5-s − 1.28·7-s + 3.06·8-s + (0.610 − 1.05i)10-s − 0.285·11-s + (2.5 − 4.33i)13-s + (−0.785 − 1.35i)14-s + (1.36 + 2.36i)16-s + (−3.11 − 5.40i)17-s + (2.92 − 3.22i)19-s − 0.507·20-s + (−0.174 − 0.301i)22-s + (−2.61 + 4.53i)23-s + ⋯
L(s)  = 1  + (0.431 + 0.748i)2-s + (0.126 − 0.219i)4-s + (−0.223 − 0.387i)5-s − 0.485·7-s + 1.08·8-s + (0.193 − 0.334i)10-s − 0.0859·11-s + (0.693 − 1.20i)13-s + (−0.209 − 0.363i)14-s + (0.341 + 0.590i)16-s + (−0.756 − 1.30i)17-s + (0.671 − 0.740i)19-s − 0.113·20-s + (−0.0371 − 0.0643i)22-s + (−0.545 + 0.945i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90619 - 0.411980i\)
\(L(\frac12)\) \(\approx\) \(1.90619 - 0.411980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-2.92 + 3.22i)T \)
good2 \( 1 + (-0.610 - 1.05i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 0.285T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.11 + 5.40i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.61 - 4.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.642 + 1.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + (0.420 + 0.728i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.47 - 4.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.86 + 4.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.18 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.86 - 4.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.22 - 3.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.492 + 0.853i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.46 + 2.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.382 - 0.661i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.72 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.66T + 83T^{2} \)
89 \( 1 + (8.01 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.87 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.993539425413899392273216462582, −9.332753240808346536102393894109, −8.177478068371166882824326237258, −7.44262949703885652720727085889, −6.62444757653147363392905694594, −5.67071689501451141749226025390, −5.05418663496903355850203714350, −3.94483291136481067690134085594, −2.65661221440307688990045431490, −0.870998846931753068844988891859, 1.67560568202253393793191788166, 2.77413244391896021600820961797, 3.88144490551443204033296595464, 4.35297532357895347315084722523, 5.99563896104815269065165633937, 6.69299712942524643545389360517, 7.70352589496402509807520460589, 8.532010695847307877442432391412, 9.575228374586160709289287373811, 10.55779453824834216903784925188

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