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Properties

Label 2-891-11.5-c1-0-27
Degree $2$
Conductor $891$
Sign $0.353 + 0.935i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
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  + (−0.118 − 0.363i)2-s + (1.5 − 1.08i)4-s + (−0.381 + 1.17i)5-s + (0.809 − 0.587i)7-s + (−1.19 − 0.865i)8-s + 0.472·10-s + (0.309 + 3.30i)11-s + (−2 − 6.15i)13-s + (−0.309 − 0.224i)14-s + (0.972 − 2.99i)16-s + (1.5 − 4.61i)17-s + (−0.809 − 0.587i)19-s + (0.708 + 2.17i)20-s + (1.16 − 0.502i)22-s + 4.61·23-s + ⋯
L(s)  = 1  + (−0.0834 − 0.256i)2-s + (0.750 − 0.544i)4-s + (−0.170 + 0.525i)5-s + (0.305 − 0.222i)7-s + (−0.421 − 0.305i)8-s + 0.149·10-s + (0.0931 + 0.995i)11-s + (−0.554 − 1.70i)13-s + (−0.0825 − 0.0600i)14-s + (0.243 − 0.747i)16-s + (0.363 − 1.11i)17-s + (−0.185 − 0.134i)19-s + (0.158 + 0.487i)20-s + (0.247 − 0.107i)22-s + 0.962·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42649 - 0.985466i\)
\(L(\frac12)\) \(\approx\) \(1.42649 - 0.985466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-0.309 - 3.30i)T \)
good2 \( 1 + (0.118 + 0.363i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.381 - 1.17i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2 + 6.15i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.5 + 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + (-3.92 + 2.85i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.11 + 2.99i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.04 - 1.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (9.66 + 7.02i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.26 - 3.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.30 - 0.951i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.35 + 10.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + (0.899 - 2.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.118 + 0.0857i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.26 - 10.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.01 - 9.28i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (-1.85 - 5.70i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.996861797188726843156050399839, −9.521721109326923643920914031837, −8.071646132572865708333395022550, −7.29277947354118403257027143045, −6.74486597400067788504987217534, −5.51141538238887306934474887787, −4.77250637324495139066941285854, −3.16586522056781641336620936753, −2.47237024591752638492640105621, −0.907438624422263675218261456862, 1.53286433284212050318867387138, 2.80804213221680473782275548595, 3.96935134380427099913842690830, 4.99075298688906689041703894827, 6.20511897678681777528378840130, 6.77712134924508694602715257828, 7.86514450986158006043543338160, 8.559666210747858449639388021185, 9.118866171254232168618064950558, 10.39515594469527129275797863522

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