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Properties

Label 1-1960-1960.1637-r0-0-0
Degree $1$
Conductor $1960$
Sign $0.383 - 0.923i$
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  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.781 − 0.623i)13-s + (−0.974 + 0.222i)17-s − 19-s + (0.974 + 0.222i)23-s + (−0.974 − 0.222i)27-s + (−0.222 − 0.974i)29-s − 31-s + (0.433 − 0.900i)33-s + (−0.974 + 0.222i)37-s + (0.900 + 0.433i)39-s + (0.900 − 0.433i)41-s + (0.433 − 0.900i)43-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.781 − 0.623i)13-s + (−0.974 + 0.222i)17-s − 19-s + (0.974 + 0.222i)23-s + (−0.974 − 0.222i)27-s + (−0.222 − 0.974i)29-s − 31-s + (0.433 − 0.900i)33-s + (−0.974 + 0.222i)37-s + (0.900 + 0.433i)39-s + (0.900 − 0.433i)41-s + (0.433 − 0.900i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(9.10220\)
Root analytic conductor: \(9.10220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1960,\ (0:\ ),\ 0.383 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8162632654 - 0.5450614774i\)
\(L(\frac12)\) \(\approx\) \(0.8162632654 - 0.5450614774i\)
\(L(1)\) \(\approx\) \(0.9882562690 + 0.1141750350i\)
\(L(1)\) \(\approx\) \(0.9882562690 + 0.1141750350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.433 + 0.900i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + (-0.974 + 0.222i)T \)
19 \( 1 - T \)
23 \( 1 + (0.974 + 0.222i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (-0.974 - 0.222i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.781 - 0.623i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−20.13494295128216166853103190694, −19.2840794132624722356672990299, −18.74950332372664023263080426931, −17.94450985570710005534455374947, −17.52697765528070354971406736543, −16.498721145561858229805516436423, −15.68314687543236390898410706166, −14.850427620414226826296161238348, −14.2973657383930780173016576439, −13.30625223632154741375818245072, −12.89919296247686195658563392227, −12.241576302265691270490473258785, −11.130205842736531518873438302383, −10.72703704226357214319984141708, −9.328310493028207710914403608582, −8.92270714892349950912200927304, −8.076554733195422755116307364803, −7.165572890914654848469164961653, −6.70809421573319664321194154581, −5.80440090849495414164826022483, −4.73947494530717566522119091047, −3.86784999903224234042486844122, −2.784586411788123339650539502128, −2.07068477926643564743533536489, −1.22243464410645341583023529684, 0.30958569368792203730701839320, 1.92564591878975029794723910752, 2.79974938749391828793522568768, 3.63226290974260524798324002787, 4.31261330561289703375399065237, 5.35628098244329390182827909716, 5.900399313609203551595129209126, 7.030421580241983447645399249639, 8.09714201942038256322347173690, 8.64299503352409715230722601285, 9.243038557065552750947410803941, 10.3406864648816331888235004066, 10.86625896967511211125611023589, 11.32387288233836119904530338617, 12.71360373218112035147352899131, 13.31479575547839304235307511388, 13.97817288792382063314084805916, 14.900538947054552002870205204659, 15.594503867202153579653108565241, 15.91137186807868058354893849412, 16.987859397777512600011416322509, 17.44560025858696511336723576572, 18.61608014412206828979311918321, 19.13667611943075159900380661403, 19.98428096768223999468747006142

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