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Properties

Label 2-855-19.7-c1-0-4
Degree $2$
Conductor $855$
Sign $0.980 - 0.194i$
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  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (−0.5 − 0.866i)5-s − 2·7-s + (−0.999 + 1.73i)10-s − 11-s + (−1 + 1.73i)13-s + (2 + 3.46i)14-s + (1.99 + 3.46i)16-s + (1 + 1.73i)17-s + (0.5 − 4.33i)19-s + 1.99·20-s + (1 + 1.73i)22-s + (−2 + 3.46i)23-s + (−0.499 + 0.866i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.223 − 0.387i)5-s − 0.755·7-s + (−0.316 + 0.547i)10-s − 0.301·11-s + (−0.277 + 0.480i)13-s + (0.534 + 0.925i)14-s + (0.499 + 0.866i)16-s + (0.242 + 0.420i)17-s + (0.114 − 0.993i)19-s + 0.447·20-s + (0.213 + 0.369i)22-s + (−0.417 + 0.722i)23-s + (−0.0999 + 0.173i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.980 - 0.194i$
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: \((2,\ 855,\ (\ :1/2),\ 0.980 - 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454846 + 0.0446687i\)
\(L(\frac12)\) \(\approx\) \(0.454846 + 0.0446687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 4.33i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06496887997571188568372579276, −9.578585425157977158166402087983, −8.823891639794973622071575360048, −7.991302407321706792940116171683, −6.87039490964308721185378737496, −5.83913511088496611302519651540, −4.57273640999181821139680983898, −3.44140134278225894866407295581, −2.55049806438771253506578357835, −1.19421792358769823450248978076, 0.31082209352004422010129065543, 2.62829742395206648303128737856, 3.74962826770606527586403925467, 5.22894987034112835940943554929, 6.09868091765882844558306414210, 6.77707456650631560855526090043, 7.66944690655925747248321042081, 8.194598918281192397179924319906, 9.193800225628552139420118205639, 9.995493294926909727146135503914

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