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Properties

Label 1-1960-1960.1269-r0-0-0
Degree $1$
Conductor $1960$
Sign $-0.814 + 0.580i$
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  + (0.365 + 0.930i)3-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)23-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.365 + 0.930i)33-s + (0.826 + 0.563i)37-s + (−0.988 + 0.149i)39-s + (0.623 − 0.781i)41-s + (0.623 + 0.781i)43-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (−0.222 + 0.974i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.0747 − 0.997i)23-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.365 + 0.930i)33-s + (0.826 + 0.563i)37-s + (−0.988 + 0.149i)39-s + (0.623 − 0.781i)41-s + (0.623 + 0.781i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(9.10220\)
Root analytic conductor: \(9.10220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1960,\ (0:\ ),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5062366777 + 1.580998044i\)
\(L(\frac12)\) \(\approx\) \(0.5062366777 + 1.580998044i\)
\(L(1)\) \(\approx\) \(1.008546509 + 0.6270122359i\)
\(L(1)\) \(\approx\) \(1.008546509 + 0.6270122359i\)

Euler product

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

−19.840398821360111866667160170340, −19.06262868773164077407797216135, −18.064231378060857728753487842854, −17.845603109567481901095943329114, −16.89919950268845109574671533914, −16.066792142987966994900314212344, −15.19068088218448449893042427255, −14.4615884297392593828140232813, −13.68261871974628080204165452648, −13.239033226999896312896827489758, −12.3416405672405441656546346697, −11.58607012133478809879184118324, −11.068959780073547576638171653842, −9.78176758602749494532551738310, −9.134882718481967748377049716781, −8.384587709108259049390246423189, −7.48866278861615713511600577143, −7.02675023798264193477476113388, −5.97595155896891041393806106082, −5.3970941981692334993782117659, −4.15863588226131763243482229571, −3.08308611020625782723899562409, −2.62710712197462835054045490626, −1.3383073549484370873839916725, −0.56801999317073690074699998984, 1.44337306398066056455758092623, 2.32715893477561720491407733261, 3.3544660331062099452867708934, 4.25976813723478000863130908757, 4.607430661281108414673422936126, 5.8102194619933350405605262262, 6.55879852489661096492105531269, 7.56461196632998804928341032557, 8.474634747107030285001973426516, 9.08092248587760287809647744645, 9.91482255378357893240688806538, 10.392409079199061829315708843396, 11.39014042324263925056785029465, 12.0656884080685195992758832530, 12.895166655882145422300273465348, 14.00807288314674718992872318951, 14.53802111664426808375862859179, 14.9643154283215692653524095389, 16.09136050744317149593193727609, 16.448634117283688947597730964874, 17.266655354475254092658814463784, 17.98937209158289189865623715402, 19.14532490446414097088090684649, 19.55046350079912385178202200004, 20.36692311362215170253642727099

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