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Properties

Label 2-855-19.5-c1-0-21
Degree $2$
Conductor $855$
Sign $-0.00611 + 0.999i$
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  + (−0.289 + 0.105i)2-s + (−1.45 + 1.22i)4-s + (−0.766 − 0.642i)5-s + (−0.0445 − 0.0772i)7-s + (0.601 − 1.04i)8-s + (0.289 + 0.105i)10-s + (−1.68 + 2.91i)11-s + (0.0369 + 0.209i)13-s + (0.0210 + 0.0176i)14-s + (0.597 − 3.38i)16-s + (−2.36 + 0.859i)17-s + (0.949 − 4.25i)19-s + 1.90·20-s + (0.180 − 1.02i)22-s + (4.57 − 3.83i)23-s + ⋯
L(s)  = 1  + (−0.204 + 0.0744i)2-s + (−0.729 + 0.612i)4-s + (−0.342 − 0.287i)5-s + (−0.0168 − 0.0291i)7-s + (0.212 − 0.368i)8-s + (0.0915 + 0.0333i)10-s + (−0.507 + 0.879i)11-s + (0.0102 + 0.0581i)13-s + (0.00562 + 0.00471i)14-s + (0.149 − 0.846i)16-s + (−0.573 + 0.208i)17-s + (0.217 − 0.975i)19-s + 0.426·20-s + (0.0383 − 0.217i)22-s + (0.953 − 0.799i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00611 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00611 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.00611 + 0.999i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.00611 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394308 - 0.396726i\)
\(L(\frac12)\) \(\approx\) \(0.394308 - 0.396726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.949 + 4.25i)T \)
good2 \( 1 + (0.289 - 0.105i)T + (1.53 - 1.28i)T^{2} \)
7 \( 1 + (0.0445 + 0.0772i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.68 - 2.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0369 - 0.209i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.36 - 0.859i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.57 + 3.83i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.51 + 1.64i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.03 + 6.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + (-0.523 + 2.96i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.87 + 1.57i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (7.15 + 2.60i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.43 + 5.39i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-9.80 + 3.56i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.757 - 0.635i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (9.37 + 3.41i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.73 + 3.97i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.73 + 15.5i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.178 - 1.01i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.96 - 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.113 + 0.646i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (15.7 - 5.75i)T + (74.3 - 62.3i)T^{2} \)
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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

−9.731468196502238393218526475398, −9.094316412546896603567978571413, −8.332349284065711789741786042988, −7.48055159296287014729073802579, −6.81545324246205448537852005078, −5.28839179700831646097943735384, −4.56181702736753307627417473951, −3.67796919249281280424958462088, −2.32672457762750709208223952765, −0.32038522799474563673285438720, 1.32875775530913650052366698388, 2.99873980229421766307835996465, 4.04788340282886082838750585842, 5.21267962100716322811229680767, 5.83960672962021473282526207462, 7.03764493797503147616752008715, 7.995265849415517592328371642908, 8.790117618896394163193753488582, 9.488094730024727856383908970456, 10.47861149028256657799216016031

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