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Properties

Label 2-804-67.37-c1-0-8
Degree $2$
Conductor $804$
Sign $0.940 + 0.339i$
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  + 3-s + 2.90·5-s + (−0.217 + 0.376i)7-s + 9-s + (2.53 − 4.39i)11-s + (−1.58 − 2.74i)13-s + 2.90·15-s + (0.150 + 0.260i)17-s + (−1.17 − 2.02i)19-s + (−0.217 + 0.376i)21-s + (1.5 + 2.59i)23-s + 3.46·25-s + 27-s + (−2.95 + 5.11i)29-s + (0.937 − 1.62i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.30·5-s + (−0.0821 + 0.142i)7-s + 0.333·9-s + (0.765 − 1.32i)11-s + (−0.440 − 0.762i)13-s + 0.751·15-s + (0.0364 + 0.0631i)17-s + (−0.268 − 0.465i)19-s + (−0.0474 + 0.0821i)21-s + (0.312 + 0.541i)23-s + 0.693·25-s + 0.192·27-s + (−0.548 + 0.950i)29-s + (0.168 − 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32886 - 0.407634i\)
\(L(\frac12)\) \(\approx\) \(2.32886 - 0.407634i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
67 \( 1 + (-8.16 + 0.601i)T \)
good5 \( 1 - 2.90T + 5T^{2} \)
7 \( 1 + (0.217 - 0.376i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.53 + 4.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.58 + 2.74i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.150 - 0.260i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.17 + 2.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.95 - 5.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.937 + 1.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.62 - 8.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.18 - 2.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.77T + 43T^{2} \)
47 \( 1 + (-3.88 + 6.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 3.34T + 59T^{2} \)
61 \( 1 + (-3.82 - 6.62i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.57 - 6.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.188 + 0.325i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.62 - 2.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + (0.127 + 0.220i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05917551293099183634860312348, −9.302662114743735793762891040321, −8.733004445445717735343922530676, −7.75398540147672816215331786272, −6.57912075097945349259843419141, −5.88655492754599426508616034665, −4.97016155828957292396104573928, −3.49409570345034926835385128310, −2.62515591209729302214247745263, −1.30530155213276258816773888159, 1.70626605116071304402635488218, 2.39876506677986799349090875808, 3.95417035603408264771270861149, 4.83096525069595337907522281480, 6.06352786563344511178295503838, 6.81850242040286476980113325583, 7.66230964321839004691689890792, 8.886854641333555134699757831451, 9.608970587551151609992614288386, 9.891247376291077499192964837682

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