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Properties

Label 2-828-23.4-c1-0-1
Degree $2$
Conductor $828$
Sign $0.952 - 0.304i$
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  + (−3.22 − 2.07i)5-s + (0.267 + 1.86i)7-s + (−0.804 + 1.76i)11-s + (−0.112 + 0.780i)13-s + (4.90 + 5.66i)17-s + (3.96 − 4.57i)19-s + (1.77 − 4.45i)23-s + (4.03 + 8.84i)25-s + (1.75 + 2.03i)29-s + (2.65 + 0.780i)31-s + (2.99 − 6.56i)35-s + (8.29 − 5.33i)37-s + (5.52 + 3.55i)41-s + (−7.72 + 2.26i)43-s + 1.75·47-s + ⋯
L(s)  = 1  + (−1.44 − 0.927i)5-s + (0.101 + 0.704i)7-s + (−0.242 + 0.531i)11-s + (−0.0311 + 0.216i)13-s + (1.18 + 1.37i)17-s + (0.909 − 1.04i)19-s + (0.370 − 0.928i)23-s + (0.807 + 1.76i)25-s + (0.326 + 0.377i)29-s + (0.477 + 0.140i)31-s + (0.507 − 1.11i)35-s + (1.36 − 0.876i)37-s + (0.863 + 0.554i)41-s + (−1.17 + 0.345i)43-s + 0.255·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15399 + 0.179840i\)
\(L(\frac12)\) \(\approx\) \(1.15399 + 0.179840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 + 4.45i)T \)
good5 \( 1 + (3.22 + 2.07i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.267 - 1.86i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.804 - 1.76i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.112 - 0.780i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-4.90 - 5.66i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-3.96 + 4.57i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-1.75 - 2.03i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-2.65 - 0.780i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-8.29 + 5.33i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-5.52 - 3.55i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (7.72 - 2.26i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 + (0.516 + 3.59i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.999 - 6.95i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (1.86 + 0.548i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-4.08 - 8.93i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-2.88 - 6.32i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (8.63 - 9.96i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.0541 + 0.376i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (2.75 - 1.77i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-4.66 + 1.36i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (7.65 + 4.91i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.25049066066569007359450231671, −9.229450503713855395368505049459, −8.467775799307375658848826351048, −7.88717796544351799477046245582, −7.01270178227399441838406688835, −5.69354043137981277251761538290, −4.79522249310122032434673322972, −4.02399855599679205480650739585, −2.79736269223143863599659273931, −1.04700804955009821439088647526, 0.78217819338393235733190692553, 3.03150069844472852810010121695, 3.51744306887987058546964816649, 4.65695844318716438271824757476, 5.82213577931411938707219403025, 7.02364198777728232568682230109, 7.71824154463611298231120138379, 8.011998879797749567252803926948, 9.507351678695440004149426521480, 10.27702134659177058070323976706

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