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Properties

Label 2-861-21.20-c1-0-100
Degree $2$
Conductor $861$
Sign $-0.550 - 0.834i$
Analytic cond. $6.87511$
Root an. cond. $2.62204$
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  − 2.32i·2-s + (−0.0978 − 1.72i)3-s − 3.41·4-s + 1.17·5-s + (−4.02 + 0.227i)6-s + (2.28 − 1.32i)7-s + 3.28i·8-s + (−2.98 + 0.338i)9-s − 2.72i·10-s − 2.67i·11-s + (0.334 + 5.90i)12-s − 5.72i·13-s + (−3.09 − 5.32i)14-s + (−0.114 − 2.02i)15-s + 0.824·16-s + 6.74·17-s + ⋯
L(s)  = 1  − 1.64i·2-s + (−0.0565 − 0.998i)3-s − 1.70·4-s + 0.523·5-s + (−1.64 + 0.0929i)6-s + (0.864 − 0.502i)7-s + 1.16i·8-s + (−0.993 + 0.112i)9-s − 0.860i·10-s − 0.806i·11-s + (0.0964 + 1.70i)12-s − 1.58i·13-s + (−0.826 − 1.42i)14-s + (−0.0295 − 0.522i)15-s + 0.206·16-s + 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.550 - 0.834i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742945 + 1.37993i\)
\(L(\frac12)\) \(\approx\) \(0.742945 + 1.37993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0978 + 1.72i)T \)
7 \( 1 + (-2.28 + 1.32i)T \)
41 \( 1 - T \)
good2 \( 1 + 2.32iT - 2T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + 5.72iT - 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 - 0.662iT - 19T^{2} \)
23 \( 1 - 0.603iT - 23T^{2} \)
29 \( 1 - 9.95iT - 29T^{2} \)
31 \( 1 - 0.464iT - 31T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 - 6.77T + 47T^{2} \)
53 \( 1 - 4.40iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 - 6.31T + 67T^{2} \)
71 \( 1 - 7.51iT - 71T^{2} \)
73 \( 1 - 9.39iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 7.30T + 83T^{2} \)
89 \( 1 - 0.109T + 89T^{2} \)
97 \( 1 + 5.59iT - 97T^{2} \)
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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

−9.982228178913617147174763193180, −8.848561125976712633753302918900, −8.071832938158280399829722112042, −7.30074997359503322239841637252, −5.76538809778942000161805799120, −5.22083391257931088073330239841, −3.59193031932063910126371350477, −2.86319113417524945900303050489, −1.58220278707781549000590726661, −0.833627351350556437318760442787, 2.16619693266902793209969384126, 4.07247923259524144838844047397, 4.74156813637332247123766821461, 5.61474909517186151985455783667, 6.16782239350703009192652747495, 7.37805222480363850127228504724, 8.070328373899637135291334331749, 9.087205016014350317767428274554, 9.464685379091084112744899957596, 10.37422060244709570459003376124

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