Properties

Label 2-855-19.11-c1-0-0
Degree $2$
Conductor $855$
Sign $-0.350 + 0.936i$
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  + (−0.809 + 1.40i)2-s + (−0.309 − 0.535i)4-s + (−0.5 + 0.866i)5-s + 0.763·7-s − 2.23·8-s + (−0.809 − 1.40i)10-s − 2·11-s + (0.618 + 1.07i)13-s + (−0.618 + 1.07i)14-s + (2.42 − 4.20i)16-s + (−1.73 + 3.00i)17-s + (−4.35 − 0.204i)19-s + 0.618·20-s + (1.61 − 2.80i)22-s + (−1.23 − 2.14i)23-s + ⋯
L(s)  = 1  + (−0.572 + 0.990i)2-s + (−0.154 − 0.267i)4-s + (−0.223 + 0.387i)5-s + 0.288·7-s − 0.790·8-s + (−0.255 − 0.443i)10-s − 0.603·11-s + (0.171 + 0.296i)13-s + (−0.165 + 0.286i)14-s + (0.606 − 1.05i)16-s + (−0.421 + 0.729i)17-s + (−0.998 − 0.0469i)19-s + 0.138·20-s + (0.344 − 0.597i)22-s + (−0.257 − 0.446i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141505 - 0.203951i\)
\(L(\frac12)\) \(\approx\) \(0.141505 - 0.203951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.204i)T \)
good2 \( 1 + (0.809 - 1.40i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 - 0.763T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-0.618 - 1.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 - 3.00i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.381 + 0.661i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.23T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 + (-3.85 + 6.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.85 - 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.881 - 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.23 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.61 - 2.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.61 + 7.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.76 + 3.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-1.76 - 3.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76772527828453078098946427036, −9.708541406198536816325608206184, −8.796456568218663949163493896983, −8.134149868197959239155425430653, −7.46149425936882518835382122063, −6.53751515112906501672725453442, −5.92240977084463553604626820082, −4.67019086659823490128490513060, −3.47773602255710980647215086111, −2.15649577034996466268618858413, 0.13745731831286773848349166266, 1.64244912668542666424442841492, 2.70697475552620787762092005647, 3.87199448299759808697495599383, 5.07313708857521987348777554546, 5.99467260526101298683004163725, 7.20029360761067328959722342019, 8.217405254825742627902501132548, 8.892993119673199644879523741957, 9.691979087996739012991985019592

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