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Properties

Label 2-804-201.5-c1-0-0
Degree $2$
Conductor $804$
Sign $-0.882 - 0.470i$
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  + (−1.57 − 0.711i)3-s + (−0.583 + 0.171i)5-s + (−0.967 − 0.838i)7-s + (1.98 + 2.24i)9-s + (1.09 − 0.322i)11-s + (−1.75 − 2.73i)13-s + (1.04 + 0.144i)15-s + (6.02 + 0.865i)17-s + (−3.63 − 4.20i)19-s + (0.931 + 2.01i)21-s + (−8.45 + 3.86i)23-s + (−3.89 + 2.50i)25-s + (−1.54 − 4.96i)27-s − 1.03i·29-s + (−4.67 + 7.26i)31-s + ⋯
L(s)  = 1  + (−0.911 − 0.410i)3-s + (−0.261 + 0.0766i)5-s + (−0.365 − 0.317i)7-s + (0.662 + 0.749i)9-s + (0.331 − 0.0973i)11-s + (−0.488 − 0.759i)13-s + (0.269 + 0.0373i)15-s + (1.46 + 0.210i)17-s + (−0.834 − 0.963i)19-s + (0.203 + 0.439i)21-s + (−1.76 + 0.805i)23-s + (−0.778 + 0.500i)25-s + (−0.296 − 0.955i)27-s − 0.192i·29-s + (−0.838 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.882 - 0.470i$
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: \((2,\ 804,\ (\ :1/2),\ -0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00399247 + 0.0159816i\)
\(L(\frac12)\) \(\approx\) \(0.00399247 + 0.0159816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 0.711i)T \)
67 \( 1 + (6.99 + 4.25i)T \)
good5 \( 1 + (0.583 - 0.171i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.967 + 0.838i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (-1.09 + 0.322i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (1.75 + 2.73i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-6.02 - 0.865i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (3.63 + 4.20i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (8.45 - 3.86i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 + 1.03iT - 29T^{2} \)
31 \( 1 + (4.67 - 7.26i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (0.975 - 6.78i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-6.37 - 0.916i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-1.82 + 0.833i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.719 - 5.00i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (4.32 - 6.73i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (2.70 - 9.20i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (12.4 - 1.78i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (2.27 + 0.667i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (8.54 + 13.2i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (3.58 + 12.1i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.31 - 0.601i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−10.38694334521349696518093131462, −10.21102239975985988256862311971, −8.933586932565978300453467369485, −7.68124160005462228391938680529, −7.32030535276120981708372367723, −6.12949101642807716839899638876, −5.52623570300391035877318371833, −4.35835213691872862739254036354, −3.24640830288397994426711788890, −1.58620043727995627292969167809, 0.009107800541335434944037993315, 1.94375150097525109058248744271, 3.71919829132666664152240989848, 4.34509641293242824321038084562, 5.64664993047530176549042743903, 6.15105471132357326770059637670, 7.22446784517147565364585419882, 8.164670096720091732497641736628, 9.328290364088302031047211571138, 9.959443111178902021115318682754

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