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Properties

Label 2-2013-2013.1280-c0-0-6
Degree $2$
Conductor $2013$
Sign $-0.606 - 0.794i$
Analytic cond. $1.00461$
Root an. cond. $1.00230$
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  + (−0.951 − 0.690i)2-s + (0.309 − 0.951i)3-s + (0.118 + 0.363i)4-s + (−0.951 + 0.690i)6-s + (−0.224 + 0.690i)8-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)11-s + 0.381·12-s + (−1.30 − 0.951i)13-s + (0.999 − 0.726i)16-s + (−1.53 + 1.11i)17-s + (0.363 + 1.11i)18-s + (−0.190 + 0.587i)19-s + (−1.11 − 0.363i)22-s − 1.17·23-s + (0.587 + 0.427i)24-s + ⋯
L(s)  = 1  + (−0.951 − 0.690i)2-s + (0.309 − 0.951i)3-s + (0.118 + 0.363i)4-s + (−0.951 + 0.690i)6-s + (−0.224 + 0.690i)8-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)11-s + 0.381·12-s + (−1.30 − 0.951i)13-s + (0.999 − 0.726i)16-s + (−1.53 + 1.11i)17-s + (0.363 + 1.11i)18-s + (−0.190 + 0.587i)19-s + (−1.11 − 0.363i)22-s − 1.17·23-s + (0.587 + 0.427i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.606 - 0.794i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ -0.606 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3284460066\)
\(L(\frac12)\) \(\approx\) \(0.3284460066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.17T + T^{2} \)
29 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - 1.90T + T^{2} \)
97 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.865104932329189360539806429887, −8.076618769911765578689848245629, −7.74753234241756270397668245073, −6.37807624109630273308822946552, −6.07494487092638656210340144130, −4.70339660913730855646977974248, −3.47951782576452356488939771412, −2.33203602162122430998700098457, −1.77545319849433863156342637341, −0.28948018696743423707878851271, 2.02047395390637128531739915075, 3.22842541792184069083393258328, 4.34357142195234502450077855190, 4.81509122245897315925370973635, 6.13961457444205815122314646852, 7.04358363676711968994866046400, 7.40480482999054068907666942671, 8.605350755599508381775289666503, 9.153251151646367720066653481341, 9.431463272000946289233737298922

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