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Properties

Label 2-891-9.4-c1-0-39
Degree $2$
Conductor $891$
Sign $-0.939 - 0.342i$
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  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (1 − 1.73i)5-s + (−2 − 3.46i)7-s − 3·8-s − 1.99·10-s + (−0.5 − 0.866i)11-s + (1 − 1.73i)13-s + (−1.99 + 3.46i)14-s + (0.500 + 0.866i)16-s − 2·17-s + (−0.999 − 1.73i)20-s + (−0.499 + 0.866i)22-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s − 1.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.447 − 0.774i)5-s + (−0.755 − 1.30i)7-s − 1.06·8-s − 0.632·10-s + (−0.150 − 0.261i)11-s + (0.277 − 0.480i)13-s + (−0.534 + 0.925i)14-s + (0.125 + 0.216i)16-s − 0.485·17-s + (−0.223 − 0.387i)20-s + (−0.106 + 0.184i)22-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173032 + 0.981313i\)
\(L(\frac12)\) \(\approx\) \(0.173032 + 0.981313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.796881391026020970327192691028, −9.153598643735392898116669754953, −8.114422174761065799486128789737, −7.04621176794054976283679933517, −6.14083880694584122465527035223, −5.34375729188104288795185261150, −4.05458031273644858220583848610, −3.01607619568294404647971627394, −1.57521238105964544316548082350, −0.51860870922335421073026234582, 2.44774566243421729489229678915, 2.84225982471357667634192984584, 4.35661074767739997125207210413, 5.90769682378417692738229776527, 6.35361440076113329356378593048, 6.96491256996299526289799693796, 8.203227840747362531681980236830, 8.754457252138944694619381754352, 9.623449107651876929411107624495, 10.38499618536219792262561892790

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