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Properties

Label 2-855-285.239-c0-0-0
Degree $2$
Conductor $855$
Sign $-0.990 - 0.140i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + i·7-s + (−1.36 − 0.366i)10-s + 1.41i·11-s + (0.866 − 0.5i)13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (0.707 − 1.22i)17-s − 19-s + (0.707 − 0.707i)20-s + (−1.73 − 1.00i)22-s + (−0.866 + 0.499i)25-s + 1.41i·26-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + i·7-s + (−1.36 − 0.366i)10-s + 1.41i·11-s + (0.866 − 0.5i)13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (0.707 − 1.22i)17-s − 19-s + (0.707 − 0.707i)20-s + (−1.73 − 1.00i)22-s + (−0.866 + 0.499i)25-s + 1.41i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.990 - 0.140i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.990 - 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6704613607\)
\(L(\frac12)\) \(\approx\) \(0.6704613607\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
19 \( 1 + T \)
good2 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.42962977440213615866889040976, −9.714263529502166216308532738357, −9.023325534312118953649715038393, −8.127858544296409656725640608433, −7.31055820282571837562642379911, −6.65468675774850185327414129291, −5.85393472339357701418459760224, −5.03225451643574385288856006494, −3.33945659040855906836525519565, −2.18249335959907441111524844653, 0.894409138103094284448744771220, 1.87410410066160454750655195747, 3.53068470354387277101090052008, 4.06879226159713545079683719338, 5.63627068413645375342238241766, 6.38692211626498296725425356518, 7.947551199842191647118411916883, 8.581970197951078498728267295972, 9.158238639823834985390617231524, 10.14673764674272547198365766836

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