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Properties

Label 2-804-67.6-c1-0-1
Degree $2$
Conductor $804$
Sign $-0.996 + 0.0776i$
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  + (−0.415 + 0.909i)3-s + (1.03 + 0.303i)5-s + (−4.65 + 0.897i)7-s + (−0.654 − 0.755i)9-s + (3.37 − 3.21i)11-s + (−0.254 + 5.33i)13-s + (−0.706 + 0.815i)15-s + (−1.13 + 0.895i)17-s + (−5.02 − 0.968i)19-s + (1.11 − 4.60i)21-s + (−3.70 + 0.354i)23-s + (−3.22 − 2.07i)25-s + (0.959 − 0.281i)27-s + (−5.06 + 8.76i)29-s + (−0.338 − 7.10i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (0.462 + 0.135i)5-s + (−1.76 + 0.339i)7-s + (−0.218 − 0.251i)9-s + (1.01 − 0.970i)11-s + (−0.0704 + 1.47i)13-s + (−0.182 + 0.210i)15-s + (−0.276 + 0.217i)17-s + (−1.15 − 0.222i)19-s + (0.244 − 1.00i)21-s + (−0.773 + 0.0738i)23-s + (−0.645 − 0.414i)25-s + (0.184 − 0.0542i)27-s + (−0.939 + 1.62i)29-s + (−0.0608 − 1.27i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.996 + 0.0776i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.996 + 0.0776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0127168 - 0.327113i\)
\(L(\frac12)\) \(\approx\) \(0.0127168 - 0.327113i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (2.15 + 7.89i)T \)
good5 \( 1 + (-1.03 - 0.303i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (4.65 - 0.897i)T + (6.49 - 2.60i)T^{2} \)
11 \( 1 + (-3.37 + 3.21i)T + (0.523 - 10.9i)T^{2} \)
13 \( 1 + (0.254 - 5.33i)T + (-12.9 - 1.23i)T^{2} \)
17 \( 1 + (1.13 - 0.895i)T + (4.00 - 16.5i)T^{2} \)
19 \( 1 + (5.02 + 0.968i)T + (17.6 + 7.06i)T^{2} \)
23 \( 1 + (3.70 - 0.354i)T + (22.5 - 4.35i)T^{2} \)
29 \( 1 + (5.06 - 8.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.338 + 7.10i)T + (-30.8 + 2.94i)T^{2} \)
37 \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.90 + 3.16i)T + (29.6 + 28.2i)T^{2} \)
43 \( 1 + (-0.383 - 2.66i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (6.39 + 8.98i)T + (-15.3 + 44.4i)T^{2} \)
53 \( 1 + (0.818 - 5.69i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-2.12 + 1.36i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (0.675 + 0.644i)T + (2.90 + 60.9i)T^{2} \)
71 \( 1 + (-7.56 - 5.94i)T + (16.7 + 68.9i)T^{2} \)
73 \( 1 + (1.07 + 1.02i)T + (3.47 + 72.9i)T^{2} \)
79 \( 1 + (7.75 - 3.99i)T + (45.8 - 64.3i)T^{2} \)
83 \( 1 + (-4.07 - 16.7i)T + (-73.7 + 38.0i)T^{2} \)
89 \( 1 + (1.53 + 3.36i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-4.41 - 7.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.59379323059433915423752654623, −9.637910425900533401510556938807, −9.269332191913720165077427748502, −8.496385972555269683596973828378, −6.67141653190557968291227830445, −6.49575331649117319604181083104, −5.63183303888210263613156281228, −4.11912672874116998472139800858, −3.48902511349123830974274968718, −2.09674125940819929016883429566, 0.15491626887966669510109335250, 1.89894418498273321836419450952, 3.20709915765488408861275620155, 4.25703258265565873746819284249, 5.70747917833127045887034336279, 6.34004496457372075476866686509, 7.03833039295417590172793809597, 8.009904584275875272967966871995, 9.182437975247464976289453366052, 9.903782638495832038557415224810

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