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Properties

Label 2-828-92.91-c1-0-47
Degree $2$
Conductor $828$
Sign $0.990 + 0.135i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
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  + (1.36 + 0.377i)2-s + (1.71 + 1.02i)4-s − 2.63i·5-s + 3.76·7-s + (1.94 + 2.04i)8-s + (0.995 − 3.59i)10-s + 0.443·11-s − 3.89·13-s + (5.12 + 1.42i)14-s + (1.88 + 3.52i)16-s − 5.41i·17-s + 0.874·19-s + (2.71 − 4.52i)20-s + (0.605 + 0.167i)22-s + (−4.40 + 1.88i)23-s + ⋯
L(s)  = 1  + (0.963 + 0.266i)2-s + (0.857 + 0.514i)4-s − 1.17i·5-s + 1.42·7-s + (0.689 + 0.724i)8-s + (0.314 − 1.13i)10-s + 0.133·11-s − 1.08·13-s + (1.37 + 0.379i)14-s + (0.470 + 0.882i)16-s − 1.31i·17-s + 0.200·19-s + (0.606 − 1.01i)20-s + (0.128 + 0.0357i)22-s + (−0.919 + 0.393i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.18067 - 0.216916i\)
\(L(\frac12)\) \(\approx\) \(3.18067 - 0.216916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.377i)T \)
3 \( 1 \)
23 \( 1 + (4.40 - 1.88i)T \)
good5 \( 1 + 2.63iT - 5T^{2} \)
7 \( 1 - 3.76T + 7T^{2} \)
11 \( 1 - 0.443T + 11T^{2} \)
13 \( 1 + 3.89T + 13T^{2} \)
17 \( 1 + 5.41iT - 17T^{2} \)
19 \( 1 - 0.874T + 19T^{2} \)
29 \( 1 - 5.34T + 29T^{2} \)
31 \( 1 - 0.598iT - 31T^{2} \)
37 \( 1 - 8.26iT - 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 - 0.586T + 43T^{2} \)
47 \( 1 + 3.89iT - 47T^{2} \)
53 \( 1 - 3.21iT - 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 - 1.61iT - 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 1.29iT - 97T^{2} \)
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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

−10.30671714749057984644715187789, −9.228046429069353258470211274935, −8.216460036174910694641383207536, −7.71454628081413467867982979182, −6.69544450865029117385299826280, −5.28152768664630888055104635402, −4.98260347846937775707229940566, −4.22792596391100328407339960437, −2.66511313502475567344758077912, −1.41558351714019901587452885287, 1.77575298825013130248228094766, 2.67139055829489731322722509075, 3.91158723185927436445894408819, 4.75995425531173228336017185391, 5.76519047869566252269565866630, 6.67065003287792229281796949340, 7.50839735532205275465901000714, 8.272214293389117522757683832754, 9.778605793724578365498130720200, 10.63114586680702314333664192549

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