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Properties

Label 1-4235-4235.2067-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.830 - 0.556i$
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  + (−0.814 − 0.580i)2-s + (0.866 + 0.5i)3-s + (0.327 + 0.945i)4-s + (−0.415 − 0.909i)6-s + (0.281 − 0.959i)8-s + (0.5 + 0.866i)9-s + (−0.189 + 0.981i)12-s + (−0.755 − 0.654i)13-s + (−0.786 + 0.618i)16-s + (0.998 − 0.0475i)17-s + (0.0950 − 0.995i)18-s + (0.0475 − 0.998i)19-s + (0.618 + 0.786i)23-s + (0.723 − 0.690i)24-s + (0.235 + 0.971i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.814 − 0.580i)2-s + (0.866 + 0.5i)3-s + (0.327 + 0.945i)4-s + (−0.415 − 0.909i)6-s + (0.281 − 0.959i)8-s + (0.5 + 0.866i)9-s + (−0.189 + 0.981i)12-s + (−0.755 − 0.654i)13-s + (−0.786 + 0.618i)16-s + (0.998 − 0.0475i)17-s + (0.0950 − 0.995i)18-s + (0.0475 − 0.998i)19-s + (0.618 + 0.786i)23-s + (0.723 − 0.690i)24-s + (0.235 + 0.971i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.830 - 0.556i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.830 - 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594452380 - 0.4848108689i\)
\(L(\frac12)\) \(\approx\) \(1.594452380 - 0.4848108689i\)
\(L(1)\) \(\approx\) \(1.028196966 - 0.1157430640i\)
\(L(1)\) \(\approx\) \(1.028196966 - 0.1157430640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.814 - 0.580i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (0.998 - 0.0475i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
23 \( 1 + (0.618 + 0.786i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.945 - 0.327i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.0950 - 0.995i)T \)
53 \( 1 + (-0.618 + 0.786i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.371 + 0.928i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.989 - 0.142i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (0.281 - 0.959i)T \)
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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

−18.57451951174889417916850669741, −17.76672411856810158863501536191, −17.165430515290890524926978735345, −16.41891445761679931995734834586, −15.8252856179001992861856598718, −14.89352439628614895001187315595, −14.48968887162173647744181346591, −14.014625810206986891858335422819, −13.1334973850089888941636174819, −12.165662651168021389934133072733, −11.78079622001025059812197038355, −10.56037382942788988398350625838, −9.92437928857601194470758212016, −9.4443959914066161598256113549, −8.58743861852922626299006806011, −8.021779975511076165299569995874, −7.51987270080213282365403699253, −6.60697747973552798313130621841, −6.26012328883870527635815742663, −5.0992772951639570739978710155, −4.359731418530578335895787034937, −3.17856577088091655989177427927, −2.48629896382754246718966915327, −1.58048569172283084697866256010, −0.9188404624560321247542060650, 0.6490489150388876114706792680, 1.67860049224319269870689990279, 2.55441829641351211760679874039, 3.12635692594544794394297801583, 3.749519398304497281433412688972, 4.7747419980584728224843160913, 5.416315822691904482192585792984, 6.90295946064784339749624043972, 7.31973115961430603010020151121, 8.15556561215388664713432235460, 8.66389043366018297013356104469, 9.45423900847000672270946988008, 9.96533647307898199801075527882, 10.55669769782858016054805663342, 11.25991450607863608109146008011, 12.21865019947449989442823741194, 12.6783485378049341860212063500, 13.6430332840835202504711117806, 14.09603861104588425931869274463, 15.25788316159646102136688632177, 15.50417595004623209658882804854, 16.299646681964353589647415834063, 17.23985112077270302399178015736, 17.44724983937547906872277809387, 18.5774417714045907741821585840

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