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Properties

Label 1-60e2-3600.1181-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.340 - 0.940i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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)7-s + (−0.406 + 0.913i)11-s + (0.406 + 0.913i)13-s + (−0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s + (−0.104 − 0.994i)23-s + (0.207 + 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.587 + 0.809i)37-s + (0.913 − 0.406i)41-s + (−0.866 + 0.5i)43-s + (0.978 − 0.207i)47-s + (−0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (0.406 + 0.913i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.406 + 0.913i)11-s + (0.406 + 0.913i)13-s + (−0.309 − 0.951i)17-s + (−0.951 + 0.309i)19-s + (−0.104 − 0.994i)23-s + (0.207 + 0.978i)29-s + (−0.978 − 0.207i)31-s + (−0.587 + 0.809i)37-s + (0.913 − 0.406i)41-s + (−0.866 + 0.5i)43-s + (0.978 − 0.207i)47-s + (−0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (0.406 + 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05026976266 - 0.07165948386i\)
\(L(\frac12)\) \(\approx\) \(0.05026976266 - 0.07165948386i\)
\(L(1)\) \(\approx\) \(0.9147224827 + 0.2146008896i\)
\(L(1)\) \(\approx\) \(0.9147224827 + 0.2146008896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (0.207 + 0.978i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (-0.406 + 0.913i)T \)
67 \( 1 + (0.207 - 0.978i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−18.742345387476515720311970583071, −17.93226327878620888866934533783, −17.30737551239154190700613153643, −16.87732621951707393089474378600, −15.85357879101366764616225439450, −15.419852007766299695490017759716, −14.537478412545692999947684608045, −13.89370192475343200226267518419, −13.12090446914309083426874774954, −12.80085608176453236142130314135, −11.59027879205611220222672503850, −10.92061744089970748240742815960, −10.58720946752003687345975907532, −9.751650796294466745211309154095, −8.64203805635514969219359434419, −8.23033868613774151127787930835, −7.502450001137056095614617296451, −6.67048997333297244245915252031, −5.76979571172533591909825797834, −5.264898760131411361015575141484, −4.04919546440889874821307976755, −3.74041211460873332806381520014, −2.64932272245198255506925808868, −1.70644181992410005706528272948, −0.78357439503027478639874033399, 0.015152842083667338135343175141, 1.405098963110571956277849707599, 2.15941234854626545936530211799, 2.74649730338818728088513313341, 3.98924061028280326580577939530, 4.66387563865960996388773042648, 5.30431538341414693201124956551, 6.22653714496965775038000365212, 6.94542250472566189413079281905, 7.661749942439643617781378381388, 8.73987803794727806966490965797, 8.90720842866092467303261923708, 9.915074375737867393330191105956, 10.70188790382963867264503190670, 11.36297068065773004182864717283, 12.20166328729293939682593533696, 12.55853053432341330946497081195, 13.5307086020246356877814865585, 14.26125485814681596671572853171, 14.94103275849850557490918427666, 15.43964000129187540927050439327, 16.330243936621972836008976888870, 16.815950385224277930315815005779, 17.87735043352051101187440935802, 18.30932584148563341887336913260

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