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Properties

Label 2-804-201.5-c1-0-14
Degree $2$
Conductor $804$
Sign $0.977 - 0.210i$
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  + (1.54 + 0.773i)3-s + (3.03 − 0.891i)5-s + (−1.25 − 1.08i)7-s + (1.80 + 2.39i)9-s + (−1.88 + 0.554i)11-s + (−0.265 − 0.413i)13-s + (5.39 + 0.968i)15-s + (2.11 + 0.304i)17-s + (4.38 + 5.06i)19-s + (−1.10 − 2.65i)21-s + (5.82 − 2.65i)23-s + (4.21 − 2.70i)25-s + (0.936 + 5.11i)27-s − 9.06i·29-s + (−3.35 + 5.22i)31-s + ⋯
L(s)  = 1  + (0.894 + 0.446i)3-s + (1.35 − 0.398i)5-s + (−0.474 − 0.410i)7-s + (0.600 + 0.799i)9-s + (−0.568 + 0.167i)11-s + (−0.0737 − 0.114i)13-s + (1.39 + 0.249i)15-s + (0.513 + 0.0737i)17-s + (1.00 + 1.16i)19-s + (−0.240 − 0.579i)21-s + (1.21 − 0.554i)23-s + (0.842 − 0.541i)25-s + (0.180 + 0.983i)27-s − 1.68i·29-s + (−0.602 + 0.937i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.51207 + 0.267371i\)
\(L(\frac12)\) \(\approx\) \(2.51207 + 0.267371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 - 0.773i)T \)
67 \( 1 + (8.02 - 1.58i)T \)
good5 \( 1 + (-3.03 + 0.891i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (1.25 + 1.08i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (1.88 - 0.554i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (0.265 + 0.413i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-2.11 - 0.304i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (-4.38 - 5.06i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-5.82 + 2.65i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 + 9.06iT - 29T^{2} \)
31 \( 1 + (3.35 - 5.22i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 + (-0.191 + 1.32i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (3.41 + 0.491i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-4.53 + 2.06i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.353 - 2.46i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.69 - 2.64i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-2.44 + 8.31i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (15.3 - 2.21i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-5.08 - 1.49i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.14 + 1.78i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-2.84 - 9.68i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (4.82 + 2.20i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−10.13124240398479180691737971860, −9.550766661245639371297010186337, −8.776931379375617995648160167536, −7.83438845239133482632453950148, −6.92105612475360182854193038114, −5.67819196171158029646879742839, −5.01720099708541455486654186848, −3.70913766243386517462664283315, −2.70679346973100403620294742626, −1.53022776613912755824529183102, 1.48152729147758408532999358212, 2.71084348926750425775584228611, 3.24073707693690773484905784992, 5.06844542286822074046971287004, 5.87850009193834806655679455282, 6.92152385070351215461824554735, 7.46991662714115268521470560825, 8.867803991046049317110717088031, 9.250726880778997802067444252004, 10.00766066518580642199207784248

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