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Properties

Label 1-1620-1620.907-r0-0-0
Degree $1$
Conductor $1620$
Sign $0.970 + 0.240i$
Analytic cond. $7.52324$
Root an. cond. $7.52324$
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  + (0.918 − 0.396i)7-s + (−0.973 + 0.230i)11-s + (−0.549 + 0.835i)13-s + (−0.342 − 0.939i)17-s + (−0.939 − 0.342i)19-s + (0.918 + 0.396i)23-s + (0.0581 + 0.998i)29-s + (0.993 − 0.116i)31-s + (−0.642 + 0.766i)37-s + (0.893 − 0.448i)41-s + (0.727 − 0.686i)43-s + (−0.116 + 0.993i)47-s + (0.686 − 0.727i)49-s + (0.866 + 0.5i)53-s + (0.973 + 0.230i)59-s + ⋯
L(s)  = 1  + (0.918 − 0.396i)7-s + (−0.973 + 0.230i)11-s + (−0.549 + 0.835i)13-s + (−0.342 − 0.939i)17-s + (−0.939 − 0.342i)19-s + (0.918 + 0.396i)23-s + (0.0581 + 0.998i)29-s + (0.993 − 0.116i)31-s + (−0.642 + 0.766i)37-s + (0.893 − 0.448i)41-s + (0.727 − 0.686i)43-s + (−0.116 + 0.993i)47-s + (0.686 − 0.727i)49-s + (0.866 + 0.5i)53-s + (0.973 + 0.230i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.970 + 0.240i$
Analytic conductor: \(7.52324\)
Root analytic conductor: \(7.52324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1620,\ (0:\ ),\ 0.970 + 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527922291 + 0.1867891349i\)
\(L(\frac12)\) \(\approx\) \(1.527922291 + 0.1867891349i\)
\(L(1)\) \(\approx\) \(1.100150849 + 0.02307874141i\)
\(L(1)\) \(\approx\) \(1.100150849 + 0.02307874141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.918 - 0.396i)T \)
11 \( 1 + (-0.973 + 0.230i)T \)
13 \( 1 + (-0.549 + 0.835i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.918 + 0.396i)T \)
29 \( 1 + (0.0581 + 0.998i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
37 \( 1 + (-0.642 + 0.766i)T \)
41 \( 1 + (0.893 - 0.448i)T \)
43 \( 1 + (0.727 - 0.686i)T \)
47 \( 1 + (-0.116 + 0.993i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.973 + 0.230i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.998 + 0.0581i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.984 - 0.173i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.448 + 0.893i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.957 - 0.286i)T \)
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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

−20.65538457408809951075772114853, −19.47426955567301956427221770918, −19.06720396620324321120053063669, −18.00992481055826446360548038724, −17.57829877693971538875643754172, −16.82623237050553945244162320140, −15.80038517340365045745032304101, −15.04544028707179915383275464682, −14.70299287367335497294846109268, −13.57491629933629186262758615325, −12.83640388859527135319862659313, −12.228842287398602486387884321134, −11.16137187509992057226844432252, −10.630840763873827196938400832097, −9.8838452648567203689773119459, −8.62913703279380415553044705590, −8.22602290314652259695704869200, −7.464853267799326492240351282956, −6.3310838917373027333325934339, −5.50671714329537644016177925651, −4.818788408169371174234303734850, −3.910619262345745596696829902758, −2.642335615337340831405930427847, −2.11287487933874177522108447513, −0.730783871071451219376087973, 0.879643733306199459845047041665, 2.08974736498750349251574643347, 2.75497252448978453961530776739, 4.10363644020118788429114758587, 4.8202792006372309520698472089, 5.38509279832382800956752724085, 6.799318026573019568528180001856, 7.233125785011460710314568225252, 8.17864532588180315103088007061, 8.93536923421419172907706711603, 9.82068962483913797922058463347, 10.752462216633123798043676692524, 11.24866544143293282669822011528, 12.14780519732109729600962021441, 12.98002494183246932059515555899, 13.82057900760950063198166762775, 14.394290753713157486478401497980, 15.28917483392095699851904985394, 15.89787530576122959780001564313, 16.95031630122701264602601695656, 17.42935477266427858739107504091, 18.22009144772423701958628459118, 18.975533716513028538552311485337, 19.70354254458469100854646751475, 20.74570415358519200954613971692

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