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Properties

Label 2-855-19.7-c1-0-23
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

\(L(1)\) \(\approx\) \(2.45369 + 0.886727i\)
\(L(\frac12)\) \(\approx\) \(2.45369 + 0.886727i\)
\(L(\frac{3}{2})\) 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

−10.50901522488210511181969847446, −9.382105018787170376431561238790, −8.255985609037920628423766607912, −7.61511259678587699150971279370, −6.78707548120050567214827755385, −5.82641339193307594593016605934, −5.13669140145518256258749056570, −4.29332051233834887107013080653, −2.75811798368891239347133007553, −1.40113092110591184169943706592, 1.56804644949140658778100805133, 2.24901109386069868055860423454, 3.86522878473372535237898889242, 4.38725548978236808850138250324, 5.44565860386466715024887462457, 6.55871362094651487045288013073, 7.76047798006082227572868905774, 8.330446420525586802329391208980, 9.266707260091059048172140654123, 10.51679808312967824655291022663

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