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Properties

Label 2-1011-1011.713-c0-0-0
Degree $2$
Conductor $1011$
Sign $-0.855 - 0.518i$
Analytic cond. $0.504554$
Root an. cond. $0.710320$
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  + (−0.330 − 0.943i)3-s + (−0.900 − 0.433i)4-s + (−1.05 − 1.68i)7-s + (−0.781 + 0.623i)9-s + (−0.111 + 0.993i)12-s + (−0.881 + 1.10i)13-s + (0.623 + 0.781i)16-s + (0.953 + 0.162i)19-s + (−1.23 + 1.55i)21-s + (−0.846 + 0.532i)25-s + (0.846 + 0.532i)27-s + (0.222 + 1.97i)28-s + (−1.08 − 0.771i)31-s + (0.974 − 0.222i)36-s + (−0.781 − 1.62i)37-s + ⋯
L(s)  = 1  + (−0.330 − 0.943i)3-s + (−0.900 − 0.433i)4-s + (−1.05 − 1.68i)7-s + (−0.781 + 0.623i)9-s + (−0.111 + 0.993i)12-s + (−0.881 + 1.10i)13-s + (0.623 + 0.781i)16-s + (0.953 + 0.162i)19-s + (−1.23 + 1.55i)21-s + (−0.846 + 0.532i)25-s + (0.846 + 0.532i)27-s + (0.222 + 1.97i)28-s + (−1.08 − 0.771i)31-s + (0.974 − 0.222i)36-s + (−0.781 − 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1011\)    =    \(3 \cdot 337\)
Sign: $-0.855 - 0.518i$
Analytic conductor: \(0.504554\)
Root analytic conductor: \(0.710320\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1011} (713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1011,\ (\ :0),\ -0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2582906903\)
\(L(\frac12)\) \(\approx\) \(0.2582906903\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.330 + 0.943i)T \)
337 \( 1 + (0.846 - 0.532i)T \)
good2 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.846 - 0.532i)T^{2} \)
7 \( 1 + (1.05 + 1.68i)T + (-0.433 + 0.900i)T^{2} \)
11 \( 1 + (-0.943 + 0.330i)T^{2} \)
13 \( 1 + (0.881 - 1.10i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.532 + 0.846i)T^{2} \)
19 \( 1 + (-0.953 - 0.162i)T + (0.943 + 0.330i)T^{2} \)
23 \( 1 + (-0.846 + 0.532i)T^{2} \)
29 \( 1 + (0.846 - 0.532i)T^{2} \)
31 \( 1 + (1.08 + 0.771i)T + (0.330 + 0.943i)T^{2} \)
37 \( 1 + (0.781 + 1.62i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (-0.146 + 0.420i)T + (-0.781 - 0.623i)T^{2} \)
47 \( 1 + (0.781 - 0.623i)T^{2} \)
53 \( 1 + (-0.846 - 0.532i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.273 + 1.60i)T + (-0.943 + 0.330i)T^{2} \)
67 \( 1 + (1.53 + 0.846i)T + (0.532 + 0.846i)T^{2} \)
71 \( 1 + (-0.943 + 0.330i)T^{2} \)
73 \( 1 + (1.96 + 0.334i)T + (0.943 + 0.330i)T^{2} \)
79 \( 1 + (0.236 - 1.03i)T + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.111 - 0.993i)T^{2} \)
89 \( 1 + (-0.846 + 0.532i)T^{2} \)
97 \( 1 + (-1.77 - 0.511i)T + (0.846 + 0.532i)T^{2} \)
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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

−9.624276477544136878174312304723, −9.082781527681954817896707597683, −7.58506489949630891078779259440, −7.31831593871717517434762813877, −6.34252289005868955823787138705, −5.44221515454304372583267112867, −4.33831344771279506931010362302, −3.46970084896247256109066106580, −1.70132862374464782947573551335, −0.25307577289224940629787861571, 2.86587006571829372914876522942, 3.32863245215199060172367933389, 4.68926537617563333965362476607, 5.44934468185896607663354890928, 5.99085359086798415641673710746, 7.40662513658010688927039926817, 8.593510056865305736179954069725, 8.990702614449070475259550875039, 9.909399530872149201171985300224, 10.14443203789045984608994129517

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