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Properties

Label 1-4235-4235.2407-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.875 + 0.483i$
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  + (−0.491 − 0.870i)2-s + (0.951 + 0.309i)3-s + (−0.516 + 0.856i)4-s + (−0.198 − 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.809 + 0.587i)9-s + (−0.755 + 0.654i)12-s + (−0.996 + 0.0855i)13-s + (−0.466 − 0.884i)16-s + (0.0570 + 0.998i)17-s + (0.113 − 0.993i)18-s + (−0.254 − 0.967i)19-s + (0.989 + 0.142i)23-s + (0.941 + 0.336i)24-s + (0.564 + 0.825i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.491 − 0.870i)2-s + (0.951 + 0.309i)3-s + (−0.516 + 0.856i)4-s + (−0.198 − 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.809 + 0.587i)9-s + (−0.755 + 0.654i)12-s + (−0.996 + 0.0855i)13-s + (−0.466 − 0.884i)16-s + (0.0570 + 0.998i)17-s + (0.113 − 0.993i)18-s + (−0.254 − 0.967i)19-s + (0.989 + 0.142i)23-s + (0.941 + 0.336i)24-s + (0.564 + 0.825i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536736251 + 0.3958682857i\)
\(L(\frac12)\) \(\approx\) \(1.536736251 + 0.3958682857i\)
\(L(1)\) \(\approx\) \(1.059106326 - 0.1271169432i\)
\(L(1)\) \(\approx\) \(1.059106326 - 0.1271169432i\)

Euler product

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

−18.55755133217408259895189067387, −17.61735125715907826671600806503, −16.94799806590711903695521393935, −16.29522688564828652008939817144, −15.54043669439855229391844839877, −14.86596744357315504387711402579, −14.40553000492324039098593007143, −13.87798171895482613809865012706, −12.94808845448792459804188019180, −12.51432173221711265725038796138, −11.39387363508882376937469018047, −10.48880639370252028078214016489, −9.71437833688047851980387851485, −9.27398753256300209354837622162, −8.589418481080948109396250816088, −7.80074513224255668198196300638, −7.266824158090872828466795349798, −6.79154478542006256935053824206, −5.744489332378334118683575874810, −5.02282637800731669944012008369, −4.20351278451100590628897525750, −3.28826825106651316369590090316, −2.33434667506438097811522333435, −1.56549074340934579980546461629, −0.5151559042577551623500346048, 0.959367794698652824734520624330, 1.995270216228147834713912190196, 2.50535118772454285312820063972, 3.27730859792817901714082458267, 4.087318651059427688819960923872, 4.62844279306128860596449727640, 5.575100015350455853367213144305, 6.983843775840146189239654763343, 7.47527374254580497396156788928, 8.20902171707408357747046015856, 9.01455865506706956378430455166, 9.360825856626008514214664319131, 10.12760172279530671979837790923, 10.799146938394098872021090876465, 11.4032951443519687474991109391, 12.384290305246116977592746770797, 13.0855934675355061582591933660, 13.33278823037247796294781131588, 14.570421829122854115301467998631, 14.76193950414887586556165880537, 15.72136215835096483204075300179, 16.58468457323828720351109356683, 17.117528744888326309535529980903, 17.876477144888542284231698218616, 18.63334027705869652342473777850

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