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Properties

Label 1-3636-3636.3503-r1-0-0
Degree $1$
Conductor $3636$
Sign $0.436 + 0.899i$
Analytic cond. $390.742$
Root an. cond. $390.742$
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  + (0.978 + 0.207i)5-s + (0.743 + 0.669i)7-s + (0.406 − 0.913i)11-s + (−0.669 − 0.743i)13-s + 17-s + (−0.309 + 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.913 + 0.406i)25-s + (0.207 + 0.978i)29-s + (0.978 + 0.207i)31-s + (0.587 + 0.809i)35-s + (0.309 − 0.951i)37-s + (−0.866 − 0.5i)41-s + (−0.104 + 0.994i)43-s + (−0.913 − 0.406i)47-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)5-s + (0.743 + 0.669i)7-s + (0.406 − 0.913i)11-s + (−0.669 − 0.743i)13-s + 17-s + (−0.309 + 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.913 + 0.406i)25-s + (0.207 + 0.978i)29-s + (0.978 + 0.207i)31-s + (0.587 + 0.809i)35-s + (0.309 − 0.951i)37-s + (−0.866 − 0.5i)41-s + (−0.104 + 0.994i)43-s + (−0.913 − 0.406i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3636\)    =    \(2^{2} \cdot 3^{2} \cdot 101\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(390.742\)
Root analytic conductor: \(390.742\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3636} (3503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3636,\ (1:\ ),\ 0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.727931917 + 1.709256922i\)
\(L(\frac12)\) \(\approx\) \(2.727931917 + 1.709256922i\)
\(L(1)\) \(\approx\) \(1.415659113 + 0.1865764435i\)
\(L(1)\) \(\approx\) \(1.415659113 + 0.1865764435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
101 \( 1 \)
good5 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.207 + 0.978i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.104 + 0.994i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−18.19371215148625330018399287453, −17.49871631255187074595979213119, −17.15364286417657707678309491304, −16.635143151966670600560937056487, −15.55084116364223638573385149570, −14.770866264573367925211498277919, −14.25307616757919619451772385070, −13.58759056122674184257201365483, −13.04374643626797735948293869758, −11.858084686299257938962874444576, −11.742488096531545404748537287520, −10.53528199942353229587839216296, −9.80403464798928789463561333055, −9.60538442516478536751146624122, −8.492296087184026327370148182713, −7.79028477077575139903453093075, −6.93882419684873494881089760423, −6.41536814983732397777724404419, −5.35152182221082798956357311382, −4.71711518135366949517085576657, −4.17123919253974468376572055838, −2.97614687971686221330222283793, −1.96192595782891922396260125394, −1.55548416270958780031965275449, −0.49006022086757917579442074466, 0.92502803425126232459528771113, 1.63984629588741573399727262763, 2.573933862583687534245672214909, 3.16071600979056522526043586846, 4.26854897701591541720920345954, 5.28839788560260398831091842417, 5.69060324307265396250681686010, 6.34178717898709432860052242191, 7.325120771028854943568364088237, 8.27769001613106775768473164808, 8.62265709861635104156157345150, 9.6362000590371539353493138582, 10.26716514423815907900800169770, 10.80943894208440466578366898965, 11.85631205224372486158185908560, 12.28741614859768890042074814716, 13.11691510286371807343337981712, 13.98287428653228556049051420061, 14.63019031591318892006233332730, 14.76633916151063971187087648036, 16.05225646636189922860134973916, 16.59555024724096551719701231906, 17.35630360127524840097157397251, 17.926299578692194516283737997868, 18.58361596363104367638867941430

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