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Properties

Label 1-4235-4235.3779-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.998 - 0.0493i$
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  + (−0.870 − 0.491i)2-s + (0.309 + 0.951i)3-s + (0.516 + 0.856i)4-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)12-s + (−0.0855 + 0.996i)13-s + (−0.466 + 0.884i)16-s + (0.998 + 0.0570i)17-s + (0.993 − 0.113i)18-s + (−0.254 + 0.967i)19-s + (0.142 + 0.989i)23-s + (0.941 − 0.336i)24-s + (0.564 − 0.825i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.870 − 0.491i)2-s + (0.309 + 0.951i)3-s + (0.516 + 0.856i)4-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)12-s + (−0.0855 + 0.996i)13-s + (−0.466 + 0.884i)16-s + (0.998 + 0.0570i)17-s + (0.993 − 0.113i)18-s + (−0.254 + 0.967i)19-s + (0.142 + 0.989i)23-s + (0.941 − 0.336i)24-s + (0.564 − 0.825i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.998 - 0.0493i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.998 - 0.0493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01697200677 + 0.6879502045i\)
\(L(\frac12)\) \(\approx\) \(0.01697200677 + 0.6879502045i\)
\(L(1)\) \(\approx\) \(0.6539967270 + 0.2707735209i\)
\(L(1)\) \(\approx\) \(0.6539967270 + 0.2707735209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.870 - 0.491i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.0855 + 0.996i)T \)
17 \( 1 + (0.998 + 0.0570i)T \)
19 \( 1 + (-0.254 + 0.967i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.774 + 0.633i)T \)
31 \( 1 + (-0.974 - 0.226i)T \)
37 \( 1 + (0.921 - 0.389i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.993 + 0.113i)T \)
53 \( 1 + (0.466 + 0.884i)T \)
59 \( 1 + (0.736 + 0.676i)T \)
61 \( 1 + (-0.870 + 0.491i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (0.985 + 0.170i)T \)
79 \( 1 + (-0.610 + 0.791i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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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

−18.18979259216650342189612750206, −17.31194960383081789755587287612, −16.966954649321746735935631565219, −16.12644927204330582200574619684, −15.192134552146033650713455185356, −14.79972515476528579721498788568, −14.1165398694074501161695038038, −13.22814767127300370682824887919, −12.691428953175549538065273115559, −11.7788778472947005286310489497, −11.17635252272723735244986594240, −10.313498027963590567620363802372, −9.673426751281666062523341363289, −8.745122526296695602971380401503, −8.36466248628772105323909598919, −7.50238243635393579505170375616, −7.146303950494524637663049734127, −6.24957445130635686679575251998, −5.65315061519688009561262400866, −4.88988967004286998872965203585, −3.49429502475921465919200540388, −2.6717644968461095910471465879, −1.97926373133099389793863649947, −0.9994006622692912662238807560, −0.27004764054187970466526860419, 1.35134205662669566953157790332, 2.03354160484617520923808837592, 3.01763672082823239549151407013, 3.69337377149903278142206852676, 4.22385304648162187714598945824, 5.34068173982784559752686680385, 6.067273952141516556580252102906, 7.20909298843608190722510371754, 7.75786251570969532729348283015, 8.55821689025329486295342906468, 9.24982249144400710826742868079, 9.68258576673071543967596600631, 10.38738552091392993700268948013, 11.03725043328627127833672530541, 11.69745264458445912952132225447, 12.31902938618701813992798900002, 13.26240407860919106358521165325, 14.05120309442573461497871309268, 14.82108784899696061118602783415, 15.38576142795468456847753185428, 16.36413087757247491452744713069, 16.70733020051644477034787302219, 17.06450394722205640128560248815, 18.3810069685469281756960611044, 18.54948609346904385257301816216

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