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Properties

Label 2-891-99.25-c1-0-17
Degree $2$
Conductor $891$
Sign $-0.177 - 0.984i$
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  + (0.0330 + 0.314i)2-s + (1.85 − 0.395i)4-s + (−0.423 + 4.03i)5-s + (2.28 + 2.53i)7-s + (0.380 + 1.17i)8-s − 1.28·10-s + (2.11 + 2.55i)11-s + (1.26 + 0.562i)13-s + (−0.721 + 0.801i)14-s + (3.11 − 1.38i)16-s + (−5.20 − 3.78i)17-s + (−1.07 − 3.31i)19-s + (0.805 + 7.66i)20-s + (−0.733 + 0.749i)22-s + (−1.86 − 3.23i)23-s + ⋯
L(s)  = 1  + (0.0233 + 0.222i)2-s + (0.929 − 0.197i)4-s + (−0.189 + 1.80i)5-s + (0.862 + 0.957i)7-s + (0.134 + 0.414i)8-s − 0.405·10-s + (0.637 + 0.770i)11-s + (0.350 + 0.155i)13-s + (−0.192 + 0.214i)14-s + (0.778 − 0.346i)16-s + (−1.26 − 0.916i)17-s + (−0.247 − 0.760i)19-s + (0.180 + 1.71i)20-s + (−0.156 + 0.159i)22-s + (−0.389 − 0.675i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.177 - 0.984i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35136 + 1.61759i\)
\(L(\frac12)\) \(\approx\) \(1.35136 + 1.61759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 2.55i)T \)
good2 \( 1 + (-0.0330 - 0.314i)T + (-1.95 + 0.415i)T^{2} \)
5 \( 1 + (0.423 - 4.03i)T + (-4.89 - 1.03i)T^{2} \)
7 \( 1 + (-2.28 - 2.53i)T + (-0.731 + 6.96i)T^{2} \)
13 \( 1 + (-1.26 - 0.562i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (5.20 + 3.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.07 + 3.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0499 + 0.0554i)T + (-3.03 + 28.8i)T^{2} \)
31 \( 1 + (3.41 + 1.52i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-6.23 + 6.92i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-0.228 + 0.395i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.14 - 0.456i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (-0.729 + 0.530i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.84 + 1.66i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-5.27 + 2.34i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-2.92 - 5.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.66 - 5.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.34 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.14 - 10.9i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (1.24 - 0.556i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-0.862 - 8.20i)T + (-94.8 + 20.1i)T^{2} \)
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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

−10.63918531720483636255908670847, −9.600737831240384404485081963836, −8.590286442268777293937036322865, −7.50974469257793913183993660902, −6.85334176598138516949563543144, −6.37460816742549666309175196742, −5.26620668212842863655888525404, −3.97709971939137004679562882907, −2.47885661618552087298043627925, −2.21058739083045129476485213465, 1.05973239339167738258236659489, 1.81713596381233582483468384259, 3.80048538623248192805025457151, 4.24416071049026031512706242629, 5.49225556717972759420145268425, 6.37213039182549431993631535542, 7.57327473417837125750663133940, 8.253031642245758603927057083607, 8.823649820796953535378813767867, 9.956578039245776300568890379081

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