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Properties

Label 1-4235-4235.3224-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.893 - 0.449i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.928 − 0.371i)2-s + (0.5 + 0.866i)3-s + (0.723 + 0.690i)4-s + (−0.142 − 0.989i)6-s + (−0.415 − 0.909i)8-s + (−0.5 + 0.866i)9-s + (−0.235 + 0.971i)12-s + (0.959 − 0.281i)13-s + (0.0475 + 0.998i)16-s + (0.327 − 0.945i)17-s + (0.786 − 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.0475 − 0.998i)23-s + (0.580 − 0.814i)24-s + (−0.995 − 0.0950i)26-s − 27-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)2-s + (0.5 + 0.866i)3-s + (0.723 + 0.690i)4-s + (−0.142 − 0.989i)6-s + (−0.415 − 0.909i)8-s + (−0.5 + 0.866i)9-s + (−0.235 + 0.971i)12-s + (0.959 − 0.281i)13-s + (0.0475 + 0.998i)16-s + (0.327 − 0.945i)17-s + (0.786 − 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.0475 − 0.998i)23-s + (0.580 − 0.814i)24-s + (−0.995 − 0.0950i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.893 - 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171579395 - 0.2779104037i\)
\(L(\frac12)\) \(\approx\) \(1.171579395 - 0.2779104037i\)
\(L(1)\) \(\approx\) \(0.8423868776 + 0.03774642430i\)
\(L(1)\) \(\approx\) \(0.8423868776 + 0.03774642430i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.928 - 0.371i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.327 - 0.945i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (-0.723 + 0.690i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.786 + 0.618i)T \)
53 \( 1 + (-0.0475 + 0.998i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.888 + 0.458i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.463213107670368116722226124835, −17.84766210876317833911807544665, −17.16664876493676636114447578544, −16.57999257567586804933250766332, −15.76667061592516680667588119195, −14.93182872117740775040493264197, −14.62484103144729773612004836137, −13.61734239846688422930634234450, −13.165059873366955406335064650806, −12.14439042841078672366856104233, −11.580946200678951696942232992579, −10.84308808603243842725755073954, −9.97143068260817482472838988374, −9.3285295070320254782912375977, −8.56415576236814769452253354781, −8.01432077459741793809575192691, −7.53504227429186230313196939966, −6.58733817580316679414952557329, −6.063303456523387759829434594022, −5.49789972818923609395513819785, −3.98905978040015015974679836148, −3.34228198033566392359541876568, −2.170915231524856391135662663781, −1.679290807606482452144600610018, −0.882700772543903219293643489563, 0.50426135837149405058639760582, 1.63165150001942840978513807334, 2.570624441316308995710601653887, 3.17125300651918550930770113843, 3.8639883111614897360593627404, 4.783785200488455409586792115451, 5.61062114306705705831432968414, 6.69822742028781649554997654789, 7.30685888302600096313446099719, 8.304514160529977061759656845573, 8.72360449863589711435101624804, 9.27626536895403292954708900487, 10.08095098431685167671712575085, 10.73942777560938944159641821400, 11.092472204422423216418698043656, 12.02288825240673161197048519793, 12.765359153984862166593772660435, 13.67310717922628519866007639132, 14.23684037742408315214632814720, 15.329616763711369803077056374090, 15.657268874652020404133732618375, 16.30296966686170206156839635769, 16.973059821332871697774434991963, 17.603203490968542356040147198712, 18.56484462185269921331125616100

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