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Properties

Label 2-804-201.200-c1-0-4
Degree $2$
Conductor $804$
Sign $0.211 - 0.977i$
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  − 1.73i·3-s + 3.46i·7-s − 2.99·9-s + 6.92i·13-s − 8·19-s + 5.99·21-s − 5·25-s + 5.19i·27-s + 10.3i·31-s + 10·37-s + 11.9·39-s − 10.3i·43-s − 4.99·49-s + 13.8i·57-s + 6.92i·61-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.30i·7-s − 0.999·9-s + 1.92i·13-s − 1.83·19-s + 1.30·21-s − 25-s + 0.999i·27-s + 1.86i·31-s + 1.64·37-s + 1.92·39-s − 1.58i·43-s − 0.714·49-s + 1.83i·57-s + 0.887i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753047 + 0.607455i\)
\(L(\frac12)\) \(\approx\) \(0.753047 + 0.607455i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
67 \( 1 + (-8 - 1.73i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 17.3iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−10.58239841310948717861466150815, −9.175268508548791978029539178588, −8.837342018888919708482711389547, −7.957295659170044267405941876662, −6.75108053456014367237310928742, −6.32190872347581637056997581125, −5.29734093237376108657972354062, −4.05669264461900204148751515068, −2.47947985292924418226681639704, −1.81510236695420132804793374849, 0.45889908113112294747661316146, 2.62395235524426326158758143451, 3.83822977238911175542549080149, 4.40960358390524532506210086186, 5.60215186231362226963792820265, 6.41887827763066451713508270105, 7.900483632880569114871967100450, 8.119684511693085169170049208515, 9.586297368693113614199599699184, 10.03707561523877332423017607821

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