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Properties

Label 2-804-804.23-c1-0-32
Degree $2$
Conductor $804$
Sign $0.984 + 0.176i$
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  + (−1.35 − 0.391i)2-s + (−0.812 − 1.52i)3-s + (1.69 + 1.06i)4-s + (0.160 − 0.0732i)5-s + (0.505 + 2.39i)6-s + (0.691 + 0.725i)7-s + (−1.88 − 2.10i)8-s + (−1.67 + 2.48i)9-s + (−0.246 + 0.0367i)10-s + (−2.26 − 0.216i)11-s + (0.251 − 3.45i)12-s + (5.00 + 0.964i)13-s + (−0.656 − 1.25i)14-s + (−0.242 − 0.185i)15-s + (1.73 + 3.60i)16-s + (0.321 − 0.623i)17-s + ⋯
L(s)  = 1  + (−0.960 − 0.276i)2-s + (−0.469 − 0.883i)3-s + (0.846 + 0.531i)4-s + (0.0717 − 0.0327i)5-s + (0.206 + 0.978i)6-s + (0.261 + 0.274i)7-s + (−0.666 − 0.745i)8-s + (−0.559 + 0.828i)9-s + (−0.0779 + 0.0116i)10-s + (−0.682 − 0.0651i)11-s + (0.0724 − 0.997i)12-s + (1.38 + 0.267i)13-s + (−0.175 − 0.335i)14-s + (−0.0625 − 0.0479i)15-s + (0.434 + 0.900i)16-s + (0.0780 − 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814190 - 0.0723561i\)
\(L(\frac12)\) \(\approx\) \(0.814190 - 0.0723561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.391i)T \)
3 \( 1 + (0.812 + 1.52i)T \)
67 \( 1 + (0.304 + 8.17i)T \)
good5 \( 1 + (-0.160 + 0.0732i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.691 - 0.725i)T + (-0.333 + 6.99i)T^{2} \)
11 \( 1 + (2.26 + 0.216i)T + (10.8 + 2.08i)T^{2} \)
13 \( 1 + (-5.00 - 0.964i)T + (12.0 + 4.83i)T^{2} \)
17 \( 1 + (-0.321 + 0.623i)T + (-9.86 - 13.8i)T^{2} \)
19 \( 1 + (2.71 - 2.84i)T + (-0.904 - 18.9i)T^{2} \)
23 \( 1 + (-1.92 + 0.769i)T + (16.6 - 15.8i)T^{2} \)
29 \( 1 + (-8.45 - 4.88i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.119 + 0.622i)T + (-28.7 + 11.5i)T^{2} \)
37 \( 1 + (-4.73 - 8.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.67 - 0.270i)T + (40.8 - 3.89i)T^{2} \)
43 \( 1 + (2.10 + 3.27i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-5.45 - 4.29i)T + (11.0 + 45.6i)T^{2} \)
53 \( 1 + (2.99 - 4.65i)T + (-22.0 - 48.2i)T^{2} \)
59 \( 1 + (1.30 + 1.50i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-5.61 + 0.535i)T + (59.8 - 11.5i)T^{2} \)
71 \( 1 + (-13.4 + 6.93i)T + (41.1 - 57.8i)T^{2} \)
73 \( 1 + (-16.0 + 1.53i)T + (71.6 - 13.8i)T^{2} \)
79 \( 1 + (-13.5 + 4.68i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (3.90 - 5.48i)T + (-27.1 - 78.4i)T^{2} \)
89 \( 1 + (-7.93 + 1.14i)T + (85.3 - 25.0i)T^{2} \)
97 \( 1 + (-0.872 - 1.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49748626221266020466228731455, −9.285096000815303384135549013419, −8.274598084210758191646323117432, −8.024219025447356324039706441789, −6.75997461186744806873502736058, −6.21791856284875596234106664262, −5.09921026264485838774126061785, −3.40481076904949156898103341124, −2.16191481722335165314297793520, −1.09816379650645005022890364260, 0.72771664563078068631668025517, 2.54480976819640471517503171320, 3.92222414911511306167277939498, 5.09861474781222062006720151941, 6.02364755033575896021298898296, 6.71680743776897767405584553088, 8.049018782560563479365580103453, 8.552034112073607792276429699952, 9.538766583609370708522020589808, 10.29628474812526642387961605413

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