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Properties

Label 2-861-21.20-c1-0-77
Degree $2$
Conductor $861$
Sign $0.974 + 0.225i$
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  + 0.517i·2-s + (1 − 1.41i)3-s + 1.73·4-s + 1.73·5-s + (0.732 + 0.517i)6-s + (1 + 2.44i)7-s + 1.93i·8-s + (−1.00 − 2.82i)9-s + 0.896i·10-s − 1.03i·11-s + (1.73 − 2.44i)12-s − 5.79i·13-s + (−1.26 + 0.517i)14-s + (1.73 − 2.44i)15-s + 2.46·16-s − 1.26·17-s + ⋯
L(s)  = 1  + 0.366i·2-s + (0.577 − 0.816i)3-s + 0.866·4-s + 0.774·5-s + (0.298 + 0.211i)6-s + (0.377 + 0.925i)7-s + 0.683i·8-s + (−0.333 − 0.942i)9-s + 0.283i·10-s − 0.312i·11-s + (0.499 − 0.707i)12-s − 1.60i·13-s + (−0.338 + 0.138i)14-s + (0.447 − 0.632i)15-s + 0.616·16-s − 0.307·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $0.974 + 0.225i$
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: \((2,\ 861,\ (\ :1/2),\ 0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68205 - 0.306927i\)
\(L(\frac12)\) \(\approx\) \(2.68205 - 0.306927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
7 \( 1 + (-1 - 2.44i)T \)
41 \( 1 - T \)
good2 \( 1 - 0.517iT - 2T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 1.03iT - 11T^{2} \)
13 \( 1 + 5.79iT - 13T^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 - 7.20iT - 23T^{2} \)
29 \( 1 - 7.20iT - 29T^{2} \)
31 \( 1 - 0.656iT - 31T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
43 \( 1 - 9.26T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + 6.83iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 6.03iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 - 9.41iT - 71T^{2} \)
73 \( 1 + 9.14iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 2.53T + 89T^{2} \)
97 \( 1 + 5.13iT - 97T^{2} \)
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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

−10.05814646093436797571950414962, −9.040026920923325086831856282744, −8.350098688718690359210121694789, −7.55073891669059659748840069568, −6.77718544248223155118719173444, −5.70124744919418315640782674466, −5.44609175089539102185047860576, −3.22053696667794130895592174434, −2.49998072328211937821743699445, −1.47464728321624426639131245811, 1.73731457309089480393973908355, 2.46582302736079626264792639623, 3.91640485315106655713933842768, 4.46590879077083181862931047450, 5.89441257001987325622642165896, 6.77321912090854487311290713492, 7.67397235525245738354266480676, 8.642848302200882407061546205956, 9.766202844346681006365925927986, 10.04047646770044952430317128039

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