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Properties

Label 1-1260-1260.607-r1-0-0
Degree $1$
Conductor $1260$
Sign $0.824 + 0.565i$
Analytic cond. $135.405$
Root an. cond. $135.405$
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  + (0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s + (0.866 − 0.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s − i·47-s + (0.866 + 0.5i)53-s − 59-s − 61-s − i·67-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s + (0.866 − 0.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s − i·47-s + (0.866 + 0.5i)53-s − 59-s − 61-s − i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(135.405\)
Root analytic conductor: \(135.405\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1260,\ (1:\ ),\ 0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.462965815 + 0.7637158636i\)
\(L(\frac12)\) \(\approx\) \(2.462965815 + 0.7637158636i\)
\(L(1)\) \(\approx\) \(1.251276196 + 0.1149063966i\)
\(L(1)\) \(\approx\) \(1.251276196 + 0.1149063966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - iT \)
71 \( 1 - T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−20.85732189655815574573287529568, −19.966756100056893462227492880972, −19.31080087507703789619078023445, −18.46012013313031384226161383332, −17.89092591025145270116896089354, −16.820603720351076666500110989531, −16.236492059573091700774054171110, −15.57024474204941450538753760257, −14.49604880859612264423486213956, −13.793410756726727916339550257740, −13.29873418094330836992121581427, −11.96469958054490253154564548341, −11.61683758343851130564841463008, −10.67258590234292703778281652222, −9.75509382912865477624423559385, −8.93201507793458369823905520994, −8.231004875439513182059886067003, −7.2296625006496471799921389128, −6.353188376196752153595546400629, −5.60874847556903128118048754554, −4.584367089283795593203272228109, −3.58652840184720450390452489896, −2.85723735010425352593854536785, −1.493812578141909761304931607804, −0.66491932882567108944946563760, 0.87778847591422283496554266781, 1.753869265921762966177080481561, 2.94391263406929386448340683617, 3.89994637786659642655458084870, 4.66075499322542993105198691569, 5.92341701562083852663190216808, 6.32409476776235536091501988913, 7.68151101216225475214979001103, 8.07847614027641362674722161586, 9.214806628994164983339665607821, 10.033515957559507443765145756179, 10.61340906736962107374516697281, 11.8363681444846826149839021525, 12.24733987992476404838456459739, 13.213056696168952465734920494739, 14.05540820025612111469940353413, 14.77621560492881989698148857592, 15.55453291439554629798880048888, 16.3745255736961983114287925116, 17.11069145482899776816495531104, 17.984140591476060167624327576840, 18.53241357713964269905352989041, 19.50701322545580219569792848527, 20.20462805260452478165401878116, 20.90425520850781557661612642867

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