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Properties

Label 2-855-95.94-c0-0-2
Degree $2$
Conductor $855$
Sign $1$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.00·4-s + 5-s − 1.41·10-s + 1.41·13-s − 0.999·16-s − 19-s + 1.00·20-s + 25-s − 2.00·26-s + 1.41·32-s − 1.41·37-s + 1.41·38-s + 49-s − 1.41·50-s + 1.41·52-s + 1.41·53-s − 1.00·64-s + 1.41·65-s + 1.41·67-s + 2.00·74-s − 1.00·76-s − 0.999·80-s − 95-s − 1.41·97-s − 1.41·98-s + 1.00·100-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s + 5-s − 1.41·10-s + 1.41·13-s − 0.999·16-s − 19-s + 1.00·20-s + 25-s − 2.00·26-s + 1.41·32-s − 1.41·37-s + 1.41·38-s + 49-s − 1.41·50-s + 1.41·52-s + 1.41·53-s − 1.00·64-s + 1.41·65-s + 1.41·67-s + 2.00·74-s − 1.00·76-s − 0.999·80-s − 95-s − 1.41·97-s − 1.41·98-s + 1.00·100-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (379, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6090074489\)
\(L(\frac12)\) \(\approx\) \(0.6090074489\)
\(L(1)\) 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 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−10.45086148821719386969462945688, −9.406427590787638692703192745680, −8.770249463260139601258757786070, −8.234173153575635250964562267969, −7.01667909629407094397518069355, −6.33609500404421226072248267111, −5.31776636594026645937298083103, −3.93930107019701489209123059048, −2.35713804217991572148981181520, −1.30458727967014322993710946472, 1.30458727967014322993710946472, 2.35713804217991572148981181520, 3.93930107019701489209123059048, 5.31776636594026645937298083103, 6.33609500404421226072248267111, 7.01667909629407094397518069355, 8.234173153575635250964562267969, 8.770249463260139601258757786070, 9.406427590787638692703192745680, 10.45086148821719386969462945688

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