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Properties

Label 2-855-19.6-c1-0-27
Degree $2$
Conductor $855$
Sign $0.288 + 0.957i$
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  + (−1.75 + 1.47i)2-s + (0.563 − 3.19i)4-s + (−0.173 − 0.984i)5-s + (1.46 − 2.54i)7-s + (1.42 + 2.46i)8-s + (1.75 + 1.47i)10-s + (−0.288 − 0.500i)11-s + (0.629 − 0.229i)13-s + (1.16 + 6.62i)14-s + (−0.0331 − 0.0120i)16-s + (0.269 − 0.226i)17-s + (−3.86 + 2.02i)19-s − 3.24·20-s + (1.24 + 0.452i)22-s + (0.715 − 4.05i)23-s + ⋯
L(s)  = 1  + (−1.24 + 1.04i)2-s + (0.281 − 1.59i)4-s + (−0.0776 − 0.440i)5-s + (0.555 − 0.962i)7-s + (0.503 + 0.872i)8-s + (0.554 + 0.465i)10-s + (−0.0870 − 0.150i)11-s + (0.174 − 0.0635i)13-s + (0.312 + 1.77i)14-s + (−0.00829 − 0.00302i)16-s + (0.0653 − 0.0548i)17-s + (−0.886 + 0.463i)19-s − 0.725·20-s + (0.264 + 0.0964i)22-s + (0.149 − 0.846i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.411796 - 0.306116i\)
\(L(\frac12)\) \(\approx\) \(0.411796 - 0.306116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.86 - 2.02i)T \)
good2 \( 1 + (1.75 - 1.47i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-1.46 + 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.288 + 0.500i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.629 + 0.229i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.269 + 0.226i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.715 + 4.05i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.30 + 1.93i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.148 - 0.257i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.30T + 37T^{2} \)
41 \( 1 + (-2.51 - 0.913i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.14 + 6.49i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.45 + 7.09i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.713 + 4.04i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.467 - 0.392i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.178 + 1.01i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.55 + 2.14i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.29 - 13.0i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (6.70 + 2.44i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.44 + 0.527i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.65 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.17 - 2.24i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-9.34 + 7.84i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.969208435558150567074862253628, −8.773223619899490169721775035871, −8.407089766878717923574054358363, −7.54586310708264237125634698209, −6.85494799496236315344658492447, −5.92352702560323419235009041928, −4.86941098777168723103909090847, −3.75155636589351647884505256838, −1.70410405572121101294815567200, −0.38369454284337775529180882121, 1.58026067301328324708511090065, 2.49115663919602160621371237086, 3.50123945331002804055497542304, 4.95453061290844127483242495856, 6.09904208128970872814781824354, 7.30813120434271315992349703150, 8.136370343797762404654083371181, 8.863065861351090286561684944706, 9.471249230294236383240978890819, 10.41360252895088066040081144211

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