Properties

Label 1-2240-2240.13-r0-0-0
Degree $1$
Conductor $2240$
Sign $-0.502 + 0.864i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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  + (−0.923 − 0.382i)3-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.382 − 0.923i)13-s − 17-s + (−0.923 − 0.382i)19-s + (−0.707 + 0.707i)23-s + (−0.382 − 0.923i)27-s + (0.382 − 0.923i)29-s + 31-s i·33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + (−0.707 − 0.707i)41-s + (−0.923 + 0.382i)43-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.382 − 0.923i)13-s − 17-s + (−0.923 − 0.382i)19-s + (−0.707 + 0.707i)23-s + (−0.382 − 0.923i)27-s + (0.382 − 0.923i)29-s + 31-s i·33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + (−0.707 − 0.707i)41-s + (−0.923 + 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03266219264 + 0.05675260485i\)
\(L(\frac12)\) \(\approx\) \(0.03266219264 + 0.05675260485i\)
\(L(1)\) \(\approx\) \(0.6193876109 - 0.1363690852i\)
\(L(1)\) \(\approx\) \(0.6193876109 - 0.1363690852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.923 - 0.382i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 - T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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

−19.44809782173273958944255087215, −18.48667993029930995650835423872, −17.982555115926211580086635689217, −17.34773064470458199082229503339, −16.45219839103375078560699332692, −16.08167942338040646511081886783, −15.14390946485930655245490443610, −14.64057223658915116199364731415, −13.50625420380986096748028006349, −12.81720472189790271705425563967, −12.03670006461922854414834547903, −11.49285549279872185559636208060, −10.49277364963778453179041611144, −10.199004853975077169314341346179, −9.163833794884595884630071924119, −8.47812261403504093781788960446, −7.33470407470776544857053968872, −6.529680311706121570052695441552, −6.12470185199792653454819608800, −4.82476416379062284546964387761, −4.53471508223419535755814486151, −3.64836064469794660653732730858, −2.30171557078708955739038795310, −1.51738968056922755095352763196, −0.02849467030628540253509140540, 0.967541350524735834722568749029, 2.07638197031154473547273643968, 3.0145543830870965711634573376, 4.1393513006644720250696143769, 4.930792768831323678365744346927, 5.82729687204052281900508420966, 6.307585515771731711831020609163, 7.13940145212899944782163645077, 8.20396282799530012862769649108, 8.53222695739494210727781882278, 9.99825864649466720556410333705, 10.39394229147265144738984044371, 11.415925880886066396626098404916, 11.61906719825149854386656342789, 12.809347766496012443759749252042, 13.31900696059628602237857283074, 13.80801931672511960296284738891, 15.22020210232023740888378910421, 15.59762668308576357697746922917, 16.419110679822911888837976420894, 17.16057990772746581314168540522, 17.81063444754092747316204184196, 18.31463102375523748419332388397, 19.230981302173942511259766924580, 19.68885627542435406292457218825

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