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Properties

Label 1-2013-2013.1064-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.848 - 0.529i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s − 20-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s − 20-s + (−0.809 − 0.587i)23-s + (−0.809 − 0.587i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3076443204 + 1.074698039i\)
\(L(\frac12)\) \(\approx\) \(-0.3076443204 + 1.074698039i\)
\(L(1)\) \(\approx\) \(1.004646598 + 0.7613137164i\)
\(L(1)\) \(\approx\) \(1.004646598 + 0.7613137164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 - T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.75399755543102105520224289076, −18.94864047851419241909123604998, −18.19510651295629868679876973967, −17.44365169006325213736897998210, −16.26914610137507144799307962659, −15.764000276184740044389820628567, −15.10151301476221174880862260574, −14.34429739439805319826824644794, −13.474032133100671234373396147526, −12.77809714211435768087669913696, −12.05497697466711924726525866214, −11.78263486003853421548636109737, −10.75996054627661841952848978630, −9.85607765475975441933372241447, −9.18050170688759850790769734761, −8.29238483074770136188299554465, −7.51790112531399298041773402594, −6.26586957872448965825129607186, −5.4439276701758988688587124878, −5.07069984779199983978523654923, −4.11650100066160441955603117492, −3.27133225423942490555411390008, −2.26838168649755413239396629500, −1.53711326288191349121434500672, −0.257246596527649825565148385559, 1.74461101665388826360911225601, 2.655091486682555367888766544265, 3.651645275799816878683085368960, 4.23174802839072531726042695255, 4.94225802894765276360395041512, 6.195080947897152678798321404883, 6.77810857860982982351580347598, 7.27703940968960225990246579843, 8.14574807358362009975568012580, 8.91755481936352363162338477615, 10.28378015301160602703095475420, 10.79737731625205735390499789317, 11.61999611045962490211804920599, 12.25821329842597628437961470656, 13.33465182372952340732034105869, 13.85438255217341620201871614892, 14.56496223562356685578906246478, 15.05424287656061319371718340440, 15.887574948187123870983773607150, 16.69118959945940346294134048495, 17.29127637050673194418488482319, 18.01906759253393223635420878746, 18.8875243263890404631459273521, 19.89046700379925684475605569179, 20.24870692761986426133005687072

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