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Properties

Label 2-804-67.25-c1-0-5
Degree $2$
Conductor $804$
Sign $0.324 - 0.946i$
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  + (−0.654 + 0.755i)3-s + (3.26 − 2.09i)5-s + (0.572 + 3.97i)7-s + (−0.142 − 0.989i)9-s + (−2.40 + 1.54i)11-s + (−2.03 + 4.46i)13-s + (−0.552 + 3.84i)15-s + (1.18 + 0.347i)17-s + (−0.964 + 6.70i)19-s + (−3.38 − 2.17i)21-s + (4.75 − 5.48i)23-s + (4.18 − 9.15i)25-s + (0.841 + 0.540i)27-s − 3.33·29-s + (0.100 + 0.221i)31-s + ⋯
L(s)  = 1  + (−0.378 + 0.436i)3-s + (1.46 − 0.938i)5-s + (0.216 + 1.50i)7-s + (−0.0474 − 0.329i)9-s + (−0.724 + 0.465i)11-s + (−0.565 + 1.23i)13-s + (−0.142 + 0.992i)15-s + (0.287 + 0.0842i)17-s + (−0.221 + 1.53i)19-s + (−0.737 − 0.474i)21-s + (0.991 − 1.14i)23-s + (0.836 − 1.83i)25-s + (0.161 + 0.104i)27-s − 0.618·29-s + (0.0181 + 0.0397i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.324 - 0.946i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.324 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31266 + 0.937848i\)
\(L(\frac12)\) \(\approx\) \(1.31266 + 0.937848i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (-3.55 + 7.37i)T \)
good5 \( 1 + (-3.26 + 2.09i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.572 - 3.97i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (2.40 - 1.54i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.03 - 4.46i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-1.18 - 0.347i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (0.964 - 6.70i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (-4.75 + 5.48i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 + 3.33T + 29T^{2} \)
31 \( 1 + (-0.100 - 0.221i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + (2.55 + 0.749i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (2.45 + 0.721i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (7.03 - 8.11i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (-7.83 + 2.30i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-1.91 - 4.19i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-10.5 - 6.76i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (-12.5 + 3.67i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (4.29 + 2.75i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.47 - 3.22i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-2.45 + 1.57i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (6.07 + 7.01i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−10.11064814461136273048914939457, −9.614126485998363891695003268821, −8.929390118811612773670102454660, −8.195908349171266504299324456855, −6.64420330541693328679995358795, −5.75924044231479153099766294115, −5.24280761474195336240049388912, −4.44517264311497465931329408417, −2.53692581147343613085762727145, −1.72949600739411962672138540450, 0.857742997645531560894187052899, 2.38485976293595455824937665587, 3.33810380038027435759177040783, 5.05747565649401067502865378981, 5.61554164459643566797241705379, 6.84565009363440414568790237881, 7.18792179880402566417571158711, 8.178431471914168040015757414602, 9.656219820655494783070722258597, 10.11991195096380878116311661456

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