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Properties

Label 2-855-19.16-c1-0-13
Degree $2$
Conductor $855$
Sign $0.687 + 0.726i$
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  + (−0.528 − 0.443i)2-s + (−0.264 − 1.50i)4-s + (−0.173 + 0.984i)5-s + (1.16 + 2.02i)7-s + (−1.21 + 2.10i)8-s + (0.528 − 0.443i)10-s + (2.28 − 3.96i)11-s + (−1.20 − 0.438i)13-s + (0.279 − 1.58i)14-s + (−1.28 + 0.468i)16-s + (0.501 + 0.420i)17-s + (3.67 − 2.34i)19-s + 1.52·20-s + (−2.96 + 1.08i)22-s + (0.966 + 5.48i)23-s + ⋯
L(s)  = 1  + (−0.373 − 0.313i)2-s + (−0.132 − 0.750i)4-s + (−0.0776 + 0.440i)5-s + (0.441 + 0.764i)7-s + (−0.429 + 0.744i)8-s + (0.167 − 0.140i)10-s + (0.690 − 1.19i)11-s + (−0.333 − 0.121i)13-s + (0.0747 − 0.424i)14-s + (−0.321 + 0.117i)16-s + (0.121 + 0.102i)17-s + (0.843 − 0.537i)19-s + 0.340·20-s + (−0.633 + 0.230i)22-s + (0.201 + 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (586, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18267 - 0.509419i\)
\(L(\frac12)\) \(\approx\) \(1.18267 - 0.509419i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-3.67 + 2.34i)T \)
good2 \( 1 + (0.528 + 0.443i)T + (0.347 + 1.96i)T^{2} \)
7 \( 1 + (-1.16 - 2.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 3.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.20 + 0.438i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.501 - 0.420i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.966 - 5.48i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.62 + 3.04i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.24 - 3.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (-8.17 + 2.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.66 + 9.44i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.84 - 4.06i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.14 + 6.50i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.51 + 3.78i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.30 + 7.38i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.0 + 8.39i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.651 - 3.69i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (7.48 - 2.72i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.92 - 2.15i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.91 + 8.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.4 - 4.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.22 - 2.70i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.988727863306258727482972972004, −9.281220306967847388038842931074, −8.612440757001868051368686471522, −7.67276261512828764464600019258, −6.43373422480499444074111220780, −5.71296683910661441753513754879, −4.89919253345266123667945770129, −3.40728586320652121218455744502, −2.31990566089050762861686528244, −0.938948313780287381781190133155, 1.11998314004766150957130750538, 2.81069538034433430544342488335, 4.28641095336767229246873242868, 4.53328866757730553151299258260, 6.15849869178836753255858541094, 7.18755714687668511208039585567, 7.66232601454049601556684258090, 8.496380902894747718581859553967, 9.465615068970515043088933835328, 9.950961732152308863798511286344

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