Properties

Label 2-943-943.367-c0-0-2
Degree $2$
Conductor $943$
Sign $-0.546 + 0.837i$
Analytic cond. $0.470618$
Root an. cond. $0.686016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 − 0.604i)2-s + (−1.32 − 1.32i)3-s + (2.28 − 1.66i)4-s + (−3.25 − 1.65i)6-s + (2.10 − 2.89i)8-s + 2.48i·9-s + (−5.21 − 0.825i)12-s + (−0.571 + 1.12i)13-s + (1.28 − 3.96i)16-s + (1.50 + 4.62i)18-s + (0.309 + 0.951i)23-s + (−6.59 + 1.04i)24-s + (−0.309 + 0.951i)25-s + (−0.385 + 2.43i)26-s + (1.96 − 1.96i)27-s + ⋯
L(s)  = 1  + (1.86 − 0.604i)2-s + (−1.32 − 1.32i)3-s + (2.28 − 1.66i)4-s + (−3.25 − 1.65i)6-s + (2.10 − 2.89i)8-s + 2.48i·9-s + (−5.21 − 0.825i)12-s + (−0.571 + 1.12i)13-s + (1.28 − 3.96i)16-s + (1.50 + 4.62i)18-s + (0.309 + 0.951i)23-s + (−6.59 + 1.04i)24-s + (−0.309 + 0.951i)25-s + (−0.385 + 2.43i)26-s + (1.96 − 1.96i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.546 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.959933495\)
\(L(\frac12)\) \(\approx\) \(1.959933495\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (-1.86 + 0.604i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (1.32 + 1.32i)T + iT^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
11 \( 1 + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.571 - 1.12i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.707 + 0.112i)T + (0.951 + 0.309i)T^{2} \)
31 \( 1 + (0.658 + 0.478i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-1.77 - 0.906i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.262 + 1.65i)T + (-0.951 + 0.309i)T^{2} \)
73 \( 1 - 1.98iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.587 + 0.809i)T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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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

−10.64582709644350761263603335528, −9.516512303351855381940013001279, −7.48573986595846427270272878087, −7.09972216039789469121181019172, −6.24515683193038874259760283159, −5.52388577418529395086012055696, −4.91879066329061849432019344409, −3.78794485907801839933770364830, −2.27492893653081643326028451790, −1.50679053844028745711695509570, 2.79332280288089708555501706200, 3.83112923950824454397766459751, 4.53638309962611994332401125096, 5.31640766189241886448822414583, 5.79842836100730898769720272284, 6.64278942039344472127891849671, 7.54514419274586473982760684799, 8.828497448222521042703766766369, 10.32441845687406434829785089765, 10.62651815735479055154699997593

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