Properties

Label 1-4235-4235.3429-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.850 + 0.525i$
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  + (−0.254 − 0.967i)2-s + (−0.809 + 0.587i)3-s + (−0.870 + 0.491i)4-s + (0.774 + 0.633i)6-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 − 0.909i)12-s + (0.736 + 0.676i)13-s + (0.516 − 0.856i)16-s + (0.0285 + 0.999i)17-s + (−0.998 − 0.0570i)18-s + (0.610 − 0.791i)19-s + (0.654 + 0.755i)23-s + (−0.985 − 0.170i)24-s + (0.466 − 0.884i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.254 − 0.967i)2-s + (−0.809 + 0.587i)3-s + (−0.870 + 0.491i)4-s + (0.774 + 0.633i)6-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 − 0.909i)12-s + (0.736 + 0.676i)13-s + (0.516 − 0.856i)16-s + (0.0285 + 0.999i)17-s + (−0.998 − 0.0570i)18-s + (0.610 − 0.791i)19-s + (0.654 + 0.755i)23-s + (−0.985 − 0.170i)24-s + (0.466 − 0.884i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05083232264 + 0.1791246952i\)
\(L(\frac12)\) \(\approx\) \(0.05083232264 + 0.1791246952i\)
\(L(1)\) \(\approx\) \(0.5987673836 - 0.07920635101i\)
\(L(1)\) \(\approx\) \(0.5987673836 - 0.07920635101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.254 - 0.967i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.736 + 0.676i)T \)
17 \( 1 + (0.0285 + 0.999i)T \)
19 \( 1 + (0.610 - 0.791i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (-0.993 + 0.113i)T \)
37 \( 1 + (-0.198 + 0.980i)T \)
41 \( 1 + (-0.362 + 0.931i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.998 + 0.0570i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (0.362 + 0.931i)T \)
61 \( 1 + (-0.254 + 0.967i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (-0.0855 - 0.996i)T \)
79 \( 1 + (-0.897 - 0.441i)T \)
83 \( 1 + (0.921 + 0.389i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.985 - 0.170i)T \)
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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

−18.06545114525487477378613571850, −17.426652109242729793127538436141, −16.69764352171594071518089284940, −16.14520290896980634092360788338, −15.71791909843787225160812830773, −14.646737669822225126621620057544, −14.16015199777178386711180490439, −13.23763546105356951447864219623, −12.881695455035142490312262430767, −12.05483107477073132378183117380, −11.06957604308168314359969972340, −10.64938310770085909108851773008, −9.717137375635107096717426646508, −9.02581037897287165652749711230, −8.11824563300325097640405188938, −7.55865447985327934341305122985, −6.93472192546704802775361446125, −6.21739083934023230564553849306, −5.46383508347458403980815683311, −5.12283458830088141897384674589, −4.09075434021859675516109871483, −3.14617891542667565312972204679, −1.81137511651349457961125268263, −1.02225645843906677177916087693, −0.077490839728826140126578727111, 1.221912275967548316584100316194, 1.7644653185301434951073309124, 3.114621209322132682010920087372, 3.64333337361218483930204032597, 4.39237627955369269555349316491, 5.11858590519798641012842900813, 5.83129555758682375500228543052, 6.751454390820827577038632046063, 7.58818767012435056827506262973, 8.64886452793794712278401799954, 9.158358114889940499626139624269, 9.82599347129123068405405892914, 10.53841025049409596265380801445, 11.20514807249886986024583492465, 11.579505717424655844585448735319, 12.27294218243367584589006107771, 13.263828480080632566582911111798, 13.460466128955404337719438568814, 14.746572390690695901344877183808, 15.18388137174719696699501288854, 16.31380634540692143896738349719, 16.61182585968304035851263666374, 17.49779473145741875086281122576, 17.88984265215688439434784921317, 18.677280697895293589796627594147

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