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Properties

Label 1-1960-1960.123-r0-0-0
Degree $1$
Conductor $1960$
Sign $-0.681 + 0.731i$
Analytic cond. $9.10220$
Root an. cond. $9.10220$
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  + (−0.680 − 0.733i)3-s + (−0.0747 + 0.997i)9-s + (0.0747 + 0.997i)11-s + (−0.433 + 0.900i)13-s + (−0.149 + 0.988i)17-s + (0.5 − 0.866i)19-s + (0.149 + 0.988i)23-s + (0.781 − 0.623i)27-s + (0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (0.680 − 0.733i)33-s + (−0.930 + 0.365i)37-s + (0.955 − 0.294i)39-s + (−0.222 − 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)3-s + (−0.0747 + 0.997i)9-s + (0.0747 + 0.997i)11-s + (−0.433 + 0.900i)13-s + (−0.149 + 0.988i)17-s + (0.5 − 0.866i)19-s + (0.149 + 0.988i)23-s + (0.781 − 0.623i)27-s + (0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (0.680 − 0.733i)33-s + (−0.930 + 0.365i)37-s + (0.955 − 0.294i)39-s + (−0.222 − 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.681 + 0.731i$
Analytic conductor: \(9.10220\)
Root analytic conductor: \(9.10220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1960,\ (0:\ ),\ -0.681 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1881141024 + 0.4323893058i\)
\(L(\frac12)\) \(\approx\) \(0.1881141024 + 0.4323893058i\)
\(L(1)\) \(\approx\) \(0.7249232574 + 0.02236344269i\)
\(L(1)\) \(\approx\) \(0.7249232574 + 0.02236344269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.680 - 0.733i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
17 \( 1 + (-0.149 + 0.988i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.149 + 0.988i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.930 + 0.365i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (0.563 + 0.826i)T \)
53 \( 1 + (-0.930 - 0.365i)T \)
59 \( 1 + (-0.955 + 0.294i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.563 + 0.826i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.433 - 0.900i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 - iT \)
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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

−19.975573471870761596302599789985, −18.78841395814373608673316043790, −18.262691266190692066192786985925, −17.46543047709838290109246859849, −16.64846126698058658979772248363, −16.22397424030584349691401409396, −15.46324317146769206540481281028, −14.67905540364981314851963499169, −13.961672835685172584112217082898, −13.035302917262431152960048542971, −12.08111157574218477071893817622, −11.644264328689474476921952230892, −10.633919068040452939913616149871, −10.24548444055726893842733013415, −9.32826739518078123034550805113, −8.56787676163663585396445703822, −7.66771407481912599176528829938, −6.63162613948238723017490163434, −5.8919393099409799869184520119, −5.16263291742731262892953616995, −4.471681092577786731615114281791, −3.35519335496190999474878382233, −2.83247722658561627715349126531, −1.21795689544136731258364606316, −0.19118837022771434708694246483, 1.38671480978080894972461024510, 1.950632149448562387727446637819, 3.04199066858380059009077419414, 4.36141768656284609371708280440, 4.92482205512831968274818748096, 5.874631342465592353436036905426, 6.78875735362882409721105881423, 7.178231665800606396965081132568, 8.09116869182146864747612552740, 9.03234079235884331042112448330, 9.9250367995605965623241882757, 10.66852499158102914584024372007, 11.59871426681011777167006005263, 12.0842328987562253576600025485, 12.7756185715588195858755116478, 13.63261647633492797878117139721, 14.18851771133113869170078435240, 15.32326207918912090180277374752, 15.81917293985654728043696192215, 16.96005303223519072474818740517, 17.39333747530524064266902740673, 17.872514392247323136244508535981, 18.91922693376531966239330564342, 19.39344858034651155975654141431, 20.04466284114174617415379526661

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