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Properties

Label 2-804-201.5-c1-0-5
Degree $2$
Conductor $804$
Sign $-0.450 - 0.892i$
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  + (−1.02 + 1.39i)3-s + (−2.48 + 0.730i)5-s + (3.64 + 3.15i)7-s + (−0.891 − 2.86i)9-s + (6.01 − 1.76i)11-s + (2.22 + 3.45i)13-s + (1.53 − 4.21i)15-s + (−0.235 − 0.0338i)17-s + (−0.749 − 0.865i)19-s + (−8.14 + 1.84i)21-s + (−1.84 + 0.840i)23-s + (1.44 − 0.926i)25-s + (4.91 + 1.69i)27-s + 0.303i·29-s + (−4.77 + 7.43i)31-s + ⋯
L(s)  = 1  + (−0.592 + 0.805i)3-s + (−1.11 + 0.326i)5-s + (1.37 + 1.19i)7-s + (−0.297 − 0.954i)9-s + (1.81 − 0.532i)11-s + (0.616 + 0.958i)13-s + (0.396 − 1.08i)15-s + (−0.0571 − 0.00821i)17-s + (−0.171 − 0.198i)19-s + (−1.77 + 0.401i)21-s + (−0.383 + 0.175i)23-s + (0.288 − 0.185i)25-s + (0.945 + 0.326i)27-s + 0.0563i·29-s + (−0.858 + 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.450 - 0.892i$
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: \((2,\ 804,\ (\ :1/2),\ -0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633042 + 1.02876i\)
\(L(\frac12)\) \(\approx\) \(0.633042 + 1.02876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 1.39i)T \)
67 \( 1 + (-6.65 + 4.76i)T \)
good5 \( 1 + (2.48 - 0.730i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-3.64 - 3.15i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (-6.01 + 1.76i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.22 - 3.45i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.235 + 0.0338i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.749 + 0.865i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (1.84 - 0.840i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 0.303iT - 29T^{2} \)
31 \( 1 + (4.77 - 7.43i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + 2.20T + 37T^{2} \)
41 \( 1 + (0.648 - 4.51i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-3.70 - 0.533i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (4.93 - 2.25i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (1.07 + 7.45i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-4.86 + 7.56i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.272 + 0.927i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (16.0 - 2.31i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-13.2 - 3.88i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.18 + 1.83i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-1.09 - 3.73i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (9.87 + 4.51i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 2.79iT - 97T^{2} \)
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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

−11.04645291630495384780296888793, −9.523240815777631268391205007178, −8.796976005896089446005992561580, −8.301142765844110738420695553150, −6.91902651668738532853082731551, −6.12507672789314646785321418082, −5.06460473034259610198887358721, −4.17428312799801568803917050573, −3.45529127826017365219714023688, −1.56485904816429524778140098829, 0.74542114171071815136227738331, 1.70411284448518778280682493276, 3.90814335936655503076730721854, 4.34135354194819220879060178244, 5.55872823668213044189266353625, 6.70579817412171237392017809130, 7.53187809370668758518135029841, 7.980321131376949102043047467792, 8.842906870433673019754533081900, 10.29844464845698918327747037135

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