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Properties

Label 1-1480-1480.1437-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.988 - 0.148i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  − i·3-s + i·7-s − 9-s + 11-s + 13-s + 17-s − i·19-s + 21-s − 23-s + i·27-s + i·29-s + i·31-s − i·33-s − i·39-s − 41-s + ⋯
L(s)  = 1  − i·3-s + i·7-s − 9-s + 11-s + 13-s + 17-s − i·19-s + 21-s − 23-s + i·27-s + i·29-s + i·31-s − i·33-s − i·39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.673139871 - 0.1250899045i\)
\(L(\frac12)\) \(\approx\) \(1.673139871 - 0.1250899045i\)
\(L(1)\) \(\approx\) \(1.152554199 - 0.1748529770i\)
\(L(1)\) \(\approx\) \(1.152554199 - 0.1748529770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−20.80765786199627937866696469349, −20.07537983075842921767119508154, −19.37002087181307746639240278282, −18.42700679322075364915616307966, −17.436733963377789742962788544263, −16.69133764775016672443681781166, −16.40620858754278490472306960134, −15.423961350759250438374394827312, −14.57217748128434206101458687309, −14.01897427689538523325474976702, −13.32183113011975226982661913708, −11.98106163160497019824806819545, −11.52419091577058804906830947575, −10.46165336949828657469732984991, −10.05837949228127125814681492102, −9.23460021397914354599667035698, −8.2837395045409491118083250768, −7.60049139926432819768084317213, −6.30356105208394247886539204129, −5.81531631745517609070059058290, −4.60428092443263457685173326806, −3.72492199626125713940233847497, −3.570307115622477740269625926825, −1.952798795361067715932259908313, −0.77755719108302954299624532370, 1.050005145954681928511395229096, 1.80002055180973310780655060610, 2.86443306485774349725897047473, 3.65040306014986621863495932650, 5.0187665544723764598266325320, 5.88787350639368584739519583278, 6.46785277556598624399221000464, 7.307789543693101675730796852821, 8.36132908616433488708765001978, 8.806023554604030811290429821051, 9.6680553546382095668374540457, 10.9713053733731408754980991651, 11.58601451849229070139488059808, 12.37138932206254157246491431428, 12.804694097523788693278769038101, 14.095580295623331322266236026337, 14.19603776254353531275128030219, 15.39156034777151688795635370025, 16.112242906046343157973474684604, 17.05540359990275948352676501335, 17.80939665295642077885558598096, 18.43885175487984879084531656581, 19.02923577367060516390853753154, 19.78560431785220038089779655959, 20.457285564164295687662067269051

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