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Properties

Label 2-855-5.4-c1-0-6
Degree $2$
Conductor $855$
Sign $0.515 + 0.856i$
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  − 2.73i·2-s − 5.48·4-s + (−1.15 − 1.91i)5-s + 2.95i·7-s + 9.52i·8-s + (−5.23 + 3.15i)10-s + 4.70·11-s + 3.69i·13-s + 8.08·14-s + 15.0·16-s + 4.86i·17-s − 19-s + (6.32 + 10.5i)20-s − 12.8i·22-s − 2.38i·23-s + ⋯
L(s)  = 1  − 1.93i·2-s − 2.74·4-s + (−0.515 − 0.856i)5-s + 1.11i·7-s + 3.36i·8-s + (−1.65 + 0.997i)10-s + 1.41·11-s + 1.02i·13-s + 2.15·14-s + 3.77·16-s + 1.18i·17-s − 0.229·19-s + (1.41 + 2.34i)20-s − 2.74i·22-s − 0.497i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846609 - 0.478450i\)
\(L(\frac12)\) \(\approx\) \(0.846609 - 0.478450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 1.91i)T \)
19 \( 1 + T \)
good2 \( 1 + 2.73iT - 2T^{2} \)
7 \( 1 - 2.95iT - 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 - 3.69iT - 13T^{2} \)
17 \( 1 - 4.86iT - 17T^{2} \)
23 \( 1 + 2.38iT - 23T^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 - 1.78T + 31T^{2} \)
37 \( 1 - 3.73iT - 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 8.82iT - 43T^{2} \)
47 \( 1 + 1.48iT - 47T^{2} \)
53 \( 1 + 1.31iT - 53T^{2} \)
59 \( 1 + 2.04T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 5.51iT - 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 5.57iT - 73T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 - 9.88iT - 97T^{2} \)
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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

−10.02288730240928257279813231282, −9.201471930115954470764530222547, −8.845091049326697426026271080318, −8.131466209666212780854155288991, −6.31721385432088137484764612091, −5.15892204082521112769885669389, −4.21815867905608372486449279415, −3.64128089501495464380021223599, −2.17307084637925482729786469552, −1.31641431888425883124885170147, 0.55504117899014873961843373517, 3.53982434130416284167420062460, 4.09712928765119557258592529118, 5.25673740718283443044442139555, 6.25773812536219592842468649578, 7.12067641240716630118594558015, 7.35210402268103194989851818202, 8.270891861692731962650344217085, 9.265878964187661037031627819632, 9.990901754523216512708953908535

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