Properties

Label 2-828-23.12-c1-0-5
Degree $2$
Conductor $828$
Sign $0.343 + 0.939i$
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  + (−3.91 + 1.15i)5-s + (−0.0825 − 0.0952i)7-s + (1.03 + 0.667i)11-s + (1.25 − 1.44i)13-s + (−0.787 − 1.72i)17-s + (0.0613 − 0.134i)19-s + (2.15 − 4.28i)23-s + (9.83 − 6.31i)25-s + (−3.11 − 6.82i)29-s + (0.335 + 2.33i)31-s + (0.433 + 0.278i)35-s + (−7.41 − 2.17i)37-s + (10.0 − 2.95i)41-s + (1.71 − 11.9i)43-s + 5.71·47-s + ⋯
L(s)  = 1  + (−1.75 + 0.514i)5-s + (−0.0311 − 0.0359i)7-s + (0.313 + 0.201i)11-s + (0.347 − 0.400i)13-s + (−0.190 − 0.418i)17-s + (0.0140 − 0.0308i)19-s + (0.448 − 0.893i)23-s + (1.96 − 1.26i)25-s + (−0.578 − 1.26i)29-s + (0.0602 + 0.419i)31-s + (0.0731 + 0.0470i)35-s + (−1.21 − 0.357i)37-s + (1.56 − 0.460i)41-s + (0.261 − 1.82i)43-s + 0.833·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679857 - 0.475025i\)
\(L(\frac12)\) \(\approx\) \(0.679857 - 0.475025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 + (-2.15 + 4.28i)T \)
good5 \( 1 + (3.91 - 1.15i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.0825 + 0.0952i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-1.03 - 0.667i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.25 + 1.44i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.787 + 1.72i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.0613 + 0.134i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (3.11 + 6.82i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.335 - 2.33i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (7.41 + 2.17i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-10.0 + 2.95i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.71 + 11.9i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 5.71T + 47T^{2} \)
53 \( 1 + (-4.37 - 5.04i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-4.98 + 5.75i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.180 + 1.25i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (8.94 - 5.75i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (7.77 - 4.99i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (5.25 - 11.4i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-2.44 + 2.81i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (7.82 + 2.29i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.00 - 6.98i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-12.7 + 3.73i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.32835315802982911516502574670, −8.980720720000046912085697668748, −8.350062423400882739839516626413, −7.34206657280805778166452034855, −6.96139066226343287179313254386, −5.64393543070951507227963522692, −4.33835628382551518342487099776, −3.74589553158048160915146806186, −2.60482380445250460518891485604, −0.47596432696421902638485467390, 1.23070716648933600391145194644, 3.19659360244088550739369449124, 4.01301543923235881196290027394, 4.80790415266157306977496919841, 6.03438874053048803199948363283, 7.21867275202786495630529322751, 7.76583704563910906482250793195, 8.750789834507264666679041279087, 9.208085887226485834817654108279, 10.64423020516331146943322662258

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