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Properties

Label 2-855-19.7-c1-0-24
Degree $2$
Conductor $855$
Sign $-0.768 + 0.639i$
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  + (−0.656 − 1.13i)2-s + (0.138 − 0.239i)4-s + (−0.5 − 0.866i)5-s + 3.11·7-s − 2.98·8-s + (−0.656 + 1.13i)10-s + 0.692·11-s + (2.07 − 3.58i)13-s + (−2.04 − 3.54i)14-s + (1.68 + 2.91i)16-s + (1.40 + 2.43i)17-s + (−1.62 − 4.04i)19-s − 0.277·20-s + (−0.454 − 0.787i)22-s + (1.51 − 2.62i)23-s + ⋯
L(s)  = 1  + (−0.464 − 0.803i)2-s + (0.0692 − 0.119i)4-s + (−0.223 − 0.387i)5-s + 1.17·7-s − 1.05·8-s + (−0.207 + 0.359i)10-s + 0.208·11-s + (0.574 − 0.995i)13-s + (−0.546 − 0.946i)14-s + (0.421 + 0.729i)16-s + (0.340 + 0.589i)17-s + (−0.373 − 0.927i)19-s − 0.0619·20-s + (−0.0969 − 0.167i)22-s + (0.315 − 0.547i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.768 + 0.639i$
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: \((2,\ 855,\ (\ :1/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438668 - 1.21385i\)
\(L(\frac12)\) \(\approx\) \(0.438668 - 1.21385i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 4.04i)T \)
good2 \( 1 + (0.656 + 1.13i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 3.11T + 7T^{2} \)
11 \( 1 - 0.692T + 11T^{2} \)
13 \( 1 + (-2.07 + 3.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.40 - 2.43i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.51 + 2.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.374T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 + (4.77 + 8.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 4.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0650 + 0.112i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.442 + 0.766i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.719 - 1.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.44 - 9.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.75 + 9.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.37 + 4.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.37 + 9.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.386T + 83T^{2} \)
89 \( 1 + (3.37 - 5.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.16 - 3.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11024524545707403973881925545, −8.876316445115328043349634518414, −8.507427775198887612825190738414, −7.53624861776174055404071265132, −6.27092677414563375656603839654, −5.37879903224547050931210501721, −4.37054696027163033609258013813, −3.09494019522953033184110928207, −1.86696174366813190151397693864, −0.77719332900058188010767277185, 1.63669639288746575279086886005, 3.14052022581388548934377660428, 4.26104081120246887782559408397, 5.42659866723643001555352881370, 6.44279410873033105364203659410, 7.17122104647591746668455081839, 7.963477526850273841014578541314, 8.578354265659024856524839818159, 9.386928380670238189291344302502, 10.48183700641666320141936176505

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