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Properties

Label 2-855-5.4-c1-0-34
Degree $2$
Conductor $855$
Sign $-0.999 + 0.0232i$
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  − 2.39i·2-s − 3.72·4-s + (2.23 − 0.0520i)5-s − 4.15i·7-s + 4.13i·8-s + (−0.124 − 5.35i)10-s + 5.71·11-s + 3.79i·13-s − 9.93·14-s + 2.44·16-s − 2.66i·17-s − 19-s + (−8.33 + 0.194i)20-s − 13.6i·22-s − 8.13i·23-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.86·4-s + (0.999 − 0.0232i)5-s − 1.56i·7-s + 1.46i·8-s + (−0.0394 − 1.69i)10-s + 1.72·11-s + 1.05i·13-s − 2.65·14-s + 0.610·16-s − 0.646i·17-s − 0.229·19-s + (−1.86 + 0.0434i)20-s − 2.91i·22-s − 1.69i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.999 + 0.0232i$
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: \((2,\ 855,\ (\ :1/2),\ -0.999 + 0.0232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0205641 - 1.76613i\)
\(L(\frac12)\) \(\approx\) \(0.0205641 - 1.76613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.0520i)T \)
19 \( 1 + T \)
good2 \( 1 + 2.39iT - 2T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 - 3.79iT - 13T^{2} \)
17 \( 1 + 2.66iT - 17T^{2} \)
23 \( 1 + 8.13iT - 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 2.31T + 31T^{2} \)
37 \( 1 - 4.68iT - 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 1.76iT - 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 - 3.04iT - 53T^{2} \)
59 \( 1 + 0.582T + 59T^{2} \)
61 \( 1 + 9.54T + 61T^{2} \)
67 \( 1 - 1.20iT - 67T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 - 3.36T + 89T^{2} \)
97 \( 1 + 0.313iT - 97T^{2} \)
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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

−10.00770718964752680305225594847, −9.266617978386094479090985705798, −8.629180733103440127829496152524, −6.92442373775096004483047330097, −6.46856839336641904046889622551, −4.63524708884199231131324471948, −4.23110941513498307700359792757, −3.10100365894853228659627898084, −1.79358863411132419659552274856, −0.965165061762396227855908382266, 1.79812051496429152689743486740, 3.42810998214085374641217001267, 4.92700633649532649304243261406, 5.71383856672579658999553822691, 6.12959932836348714288431793011, 6.92842697407291718277877524289, 8.083112437714742856944583339317, 8.916004359680621011767017351660, 9.223513366474003555966427406833, 10.17663077077124254321707202592

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