Properties

Label 2-804-12.11-c1-0-76
Degree $2$
Conductor $804$
Sign $-0.276 + 0.960i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.38 − 0.296i)2-s + (−1.71 + 0.246i)3-s + (1.82 + 0.820i)4-s − 2.85i·5-s + (2.44 + 0.168i)6-s − 2.12i·7-s + (−2.27 − 1.67i)8-s + (2.87 − 0.843i)9-s + (−0.846 + 3.94i)10-s + 5.05·11-s + (−3.32 − 0.958i)12-s + 1.45·13-s + (−0.630 + 2.93i)14-s + (0.701 + 4.89i)15-s + (2.65 + 2.99i)16-s − 0.803i·17-s + ⋯
L(s)  = 1  + (−0.977 − 0.209i)2-s + (−0.989 + 0.142i)3-s + (0.911 + 0.410i)4-s − 1.27i·5-s + (0.997 + 0.0688i)6-s − 0.802i·7-s + (−0.805 − 0.592i)8-s + (0.959 − 0.281i)9-s + (−0.267 + 1.24i)10-s + 1.52·11-s + (−0.960 − 0.276i)12-s + 0.403·13-s + (−0.168 + 0.784i)14-s + (0.181 + 1.26i)15-s + (0.663 + 0.748i)16-s − 0.194i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.276 + 0.960i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464038 - 0.616539i\)
\(L(\frac12)\) \(\approx\) \(0.464038 - 0.616539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.296i)T \)
3 \( 1 + (1.71 - 0.246i)T \)
67 \( 1 - iT \)
good5 \( 1 + 2.85iT - 5T^{2} \)
7 \( 1 + 2.12iT - 7T^{2} \)
11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 - 1.45T + 13T^{2} \)
17 \( 1 + 0.803iT - 17T^{2} \)
19 \( 1 + 3.66iT - 19T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 - 8.38iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 - 1.53T + 37T^{2} \)
41 \( 1 + 0.167iT - 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 4.56T + 59T^{2} \)
61 \( 1 + 7.45T + 61T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 0.104T + 73T^{2} \)
79 \( 1 + 6.42iT - 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 6.70iT - 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−10.03294647156070803670336434388, −8.956487172934158491820764543482, −8.827972162000425265565554759836, −7.23450384603978982578221504742, −6.82227878426309416081134028761, −5.65822841108021421354237238232, −4.56108238240381300325055068119, −3.64959728672534208159817896743, −1.45256085679136644710067946768, −0.72968328434790877161942234966, 1.36124798913679607621453340219, 2.67507057079669281330504990904, 4.12573437342788068098377341473, 5.97059138777998959408721750386, 6.06777055669600585444879977980, 6.99602554748467530385679866697, 7.78633735714630045055463868478, 8.954758110178129316993137885233, 9.743308919591234913482550137385, 10.51561843042299437451416869974

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