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Properties

Label 1-4235-4235.2897-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.739 - 0.672i$
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  + (0.389 − 0.921i)2-s + (0.587 − 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (−0.931 + 0.362i)8-s + (−0.309 − 0.951i)9-s + (−0.989 + 0.142i)12-s + (0.441 + 0.897i)13-s + (−0.0285 + 0.999i)16-s + (−0.676 + 0.736i)17-s + (−0.996 − 0.0855i)18-s + (0.198 − 0.980i)19-s + (0.281 + 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.389 − 0.921i)2-s + (0.587 − 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (−0.931 + 0.362i)8-s + (−0.309 − 0.951i)9-s + (−0.989 + 0.142i)12-s + (0.441 + 0.897i)13-s + (−0.0285 + 0.999i)16-s + (−0.676 + 0.736i)17-s + (−0.996 − 0.0855i)18-s + (0.198 − 0.980i)19-s + (0.281 + 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.739 - 0.672i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.739 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8689517085 - 2.247300079i\)
\(L(\frac12)\) \(\approx\) \(0.8689517085 - 2.247300079i\)
\(L(1)\) \(\approx\) \(1.027957140 - 1.027014830i\)
\(L(1)\) \(\approx\) \(1.027957140 - 1.027014830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.389 - 0.921i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.441 + 0.897i)T \)
17 \( 1 + (-0.676 + 0.736i)T \)
19 \( 1 + (0.198 - 0.980i)T \)
23 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (0.985 - 0.170i)T \)
37 \( 1 + (0.884 - 0.466i)T \)
41 \( 1 + (-0.974 + 0.226i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (0.996 - 0.0855i)T \)
53 \( 1 + (-0.999 + 0.0285i)T \)
59 \( 1 + (0.974 + 0.226i)T \)
61 \( 1 + (0.921 - 0.389i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (0.791 + 0.610i)T \)
79 \( 1 + (-0.774 - 0.633i)T \)
83 \( 1 + (-0.825 - 0.564i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.967 + 0.254i)T \)
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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

−18.509751875301211169842597773131, −17.81135651613469286053326934423, −17.07214953265700827867969618449, −16.393464547620678658917306747607, −15.762734452692759722375254615716, −15.395695872922560704228606197161, −14.59900091413600550612624601117, −14.059990304672060698275166414140, −13.44076900777375676565643406819, −12.77418129903185657133129265638, −11.912379317612835522667767890471, −11.046628213477595717873150448326, −10.136948372055827556782138511, −9.65848228957986568059452142038, −8.6769777776379970791432104225, −8.295638400990819175039762644443, −7.65640671112575007231463909697, −6.68899279046798291105144475492, −5.97248569218834963842119675606, −5.160819797291001553557212710769, −4.53644016989658365079869918985, −3.88435246421102641962674935987, −3.00548411627025828980532492613, −2.490938182041857191134852711623, −0.854804370904327364794067980256, 0.69881186533734364492430319694, 1.5203919252623602345938824956, 2.22366117590282794438980483458, 2.929375793782436075108210493, 3.74837120277441813696577397609, 4.40955481458276053880727571894, 5.34215833709424696538814486959, 6.31217084799368787276014759073, 6.75998820786261964011196342823, 7.77219938694238705954144171584, 8.72774169436438495800969364872, 9.01147250887632259638944821263, 9.83570578182918011406011811412, 10.73377616295523492287316579817, 11.51526375674227978375409793190, 11.88593436142535979379052444204, 12.859623602089972272784932740837, 13.232975069992062459973082783584, 13.9295444220322248501858118910, 14.36229711952562660426494821983, 15.32281741115768898905146803144, 15.73041504314134535202266830369, 17.16320537991255631114825210781, 17.58906736924237980644270129182, 18.365052110139960189302601861388

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