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Properties

Label 2-804-201.5-c1-0-13
Degree $2$
Conductor $804$
Sign $0.736 - 0.676i$
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  + (1.33 + 1.10i)3-s + (1.09 − 0.321i)5-s + (2.09 + 1.81i)7-s + (0.575 + 2.94i)9-s + (3.69 − 1.08i)11-s + (−1.65 − 2.57i)13-s + (1.81 + 0.774i)15-s + (4.10 + 0.590i)17-s + (−4.90 − 5.66i)19-s + (0.800 + 4.72i)21-s + (−1.08 + 0.496i)23-s + (−3.11 + 2.00i)25-s + (−2.47 + 4.56i)27-s + 7.40i·29-s + (0.00434 − 0.00676i)31-s + ⋯
L(s)  = 1  + (0.771 + 0.635i)3-s + (0.489 − 0.143i)5-s + (0.790 + 0.684i)7-s + (0.191 + 0.981i)9-s + (1.11 − 0.326i)11-s + (−0.458 − 0.714i)13-s + (0.468 + 0.200i)15-s + (0.995 + 0.143i)17-s + (−1.12 − 1.29i)19-s + (0.174 + 1.03i)21-s + (−0.226 + 0.103i)23-s + (−0.622 + 0.400i)25-s + (−0.475 + 0.879i)27-s + 1.37i·29-s + (0.000780 − 0.00121i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.736 - 0.676i$
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: \((2,\ 804,\ (\ :1/2),\ 0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25645 + 0.879410i\)
\(L(\frac12)\) \(\approx\) \(2.25645 + 0.879410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
67 \( 1 + (4.74 - 6.66i)T \)
good5 \( 1 + (-1.09 + 0.321i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-2.09 - 1.81i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (-3.69 + 1.08i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (1.65 + 2.57i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-4.10 - 0.590i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (4.90 + 5.66i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (1.08 - 0.496i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 7.40iT - 29T^{2} \)
31 \( 1 + (-0.00434 + 0.00676i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (-0.919 + 6.39i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (7.53 + 1.08i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-3.24 + 1.48i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.624 + 4.34i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (7.02 - 10.9i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (0.929 - 3.16i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (-11.4 + 1.64i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-11.3 - 3.32i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (0.942 + 1.46i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (2.07 + 7.08i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (5.41 + 2.47i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 6.48iT - 97T^{2} \)
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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

−10.25577893206940622684994213203, −9.348074021617422238912030826241, −8.784681042489010440270982991409, −8.080173264622361919602477418482, −7.01458775110271380785574560194, −5.70468556077861143598391193141, −5.01450490353828811249380661581, −3.92702571670779501030904622749, −2.78581416897294904847708929929, −1.67497295087316841551681621899, 1.38172292033874705172868562851, 2.20734316046450928037431142986, 3.75322271553064198127563044332, 4.47867900415948748158498415597, 6.09282549904122386646519554627, 6.67070132612045139880978661940, 7.84506247215122072697265528299, 8.135939059911045462446137305771, 9.550979843946799773257102695793, 9.777049067840222036556735543362

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