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Properties

Label 2-804-67.64-c1-0-10
Degree $2$
Conductor $804$
Sign $-0.787 + 0.615i$
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  + (0.142 − 0.989i)3-s + (−1.42 − 3.11i)5-s + (2.11 + 0.621i)7-s + (−0.959 − 0.281i)9-s + (−0.794 − 1.73i)11-s + (0.629 − 0.726i)13-s + (−3.28 + 0.964i)15-s + (5.57 − 3.58i)17-s + (−7.43 + 2.18i)19-s + (0.915 − 2.00i)21-s + (0.573 − 3.98i)23-s + (−4.40 + 5.08i)25-s + (−0.415 + 0.909i)27-s − 0.158·29-s + (−0.313 − 0.362i)31-s + ⋯
L(s)  = 1  + (0.0821 − 0.571i)3-s + (−0.636 − 1.39i)5-s + (0.799 + 0.234i)7-s + (−0.319 − 0.0939i)9-s + (−0.239 − 0.524i)11-s + (0.174 − 0.201i)13-s + (−0.848 + 0.249i)15-s + (1.35 − 0.868i)17-s + (−1.70 + 0.501i)19-s + (0.199 − 0.437i)21-s + (0.119 − 0.831i)23-s + (−0.881 + 1.01i)25-s + (−0.0799 + 0.175i)27-s − 0.0293·29-s + (−0.0563 − 0.0650i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.787 + 0.615i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.787 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397570 - 1.15415i\)
\(L(\frac12)\) \(\approx\) \(0.397570 - 1.15415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-8.18 + 0.219i)T \)
good5 \( 1 + (1.42 + 3.11i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-2.11 - 0.621i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.794 + 1.73i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.629 + 0.726i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-5.57 + 3.58i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (7.43 - 2.18i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.573 + 3.98i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 0.158T + 29T^{2} \)
31 \( 1 + (0.313 + 0.362i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 + (6.45 - 4.14i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (0.750 - 0.482i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (0.408 - 2.83i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (0.988 + 0.635i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (4.69 + 5.42i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-3.47 + 7.59i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-7.18 - 4.61i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-4.82 + 10.5i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (5.28 - 6.10i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-3.65 - 7.99i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (1.53 + 10.7i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−9.833973150319145537259268635808, −8.621416295596842209038303763109, −8.346699533123102390430826352639, −7.67886824472791203432481146486, −6.39882153476284245159610775814, −5.30609046419451966080194881188, −4.67452561285169327473251403950, −3.42274465988599039153818537074, −1.85695399594987025981668354483, −0.60528165943718322312216678765, 2.01161190194307867164030252899, 3.34005532154251405214809377840, 4.06392146964906956729964547909, 5.15183178577919909776829292859, 6.33822093115785638448294048231, 7.25178507118981423872365563174, 7.966233999191387897391368529615, 8.804988593691681439809542770809, 10.14855157399343372233952294420, 10.50545569301370836114076477204

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