Properties

Label 1-4235-4235.768-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.718 + 0.695i$
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  + (−0.244 + 0.969i)2-s + (0.743 − 0.669i)3-s + (−0.879 − 0.475i)4-s + (0.466 + 0.884i)6-s + (0.676 − 0.736i)8-s + (0.104 − 0.994i)9-s + (−0.971 + 0.235i)12-s + (0.791 − 0.610i)13-s + (0.548 + 0.836i)16-s + (−0.572 + 0.820i)17-s + (0.938 + 0.345i)18-s + (0.797 + 0.603i)19-s + (0.998 + 0.0475i)23-s + (0.00951 − 0.999i)24-s + (0.398 + 0.917i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.244 + 0.969i)2-s + (0.743 − 0.669i)3-s + (−0.879 − 0.475i)4-s + (0.466 + 0.884i)6-s + (0.676 − 0.736i)8-s + (0.104 − 0.994i)9-s + (−0.971 + 0.235i)12-s + (0.791 − 0.610i)13-s + (0.548 + 0.836i)16-s + (−0.572 + 0.820i)17-s + (0.938 + 0.345i)18-s + (0.797 + 0.603i)19-s + (0.998 + 0.0475i)23-s + (0.00951 − 0.999i)24-s + (0.398 + 0.917i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (768, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869006200 + 0.7566099489i\)
\(L(\frac12)\) \(\approx\) \(1.869006200 + 0.7566099489i\)
\(L(1)\) \(\approx\) \(1.183128766 + 0.2714982031i\)
\(L(1)\) \(\approx\) \(1.183128766 + 0.2714982031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.244 + 0.969i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (0.791 - 0.610i)T \)
17 \( 1 + (-0.572 + 0.820i)T \)
19 \( 1 + (0.797 + 0.603i)T \)
23 \( 1 + (0.998 + 0.0475i)T \)
29 \( 1 + (-0.516 + 0.856i)T \)
31 \( 1 + (0.761 - 0.647i)T \)
37 \( 1 + (0.901 + 0.432i)T \)
41 \( 1 + (-0.897 - 0.441i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (-0.938 + 0.345i)T \)
53 \( 1 + (0.836 + 0.548i)T \)
59 \( 1 + (-0.0665 + 0.997i)T \)
61 \( 1 + (-0.272 + 0.962i)T \)
67 \( 1 + (-0.618 + 0.786i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (-0.703 + 0.710i)T \)
79 \( 1 + (0.948 - 0.318i)T \)
83 \( 1 + (0.931 + 0.362i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (-0.491 - 0.870i)T \)
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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

−18.43825909188188695465577064800, −17.82869125582264041213049186895, −16.92826614662564702838140915523, −16.28287651019342998275941628912, −15.60920198477058130625632314770, −14.83974031065104554791071719814, −13.95945255554097265591349452540, −13.54832351719542609680791185280, −13.032283515222052844816243473015, −11.9236923662276166364799713355, −11.2769922771532539785850060303, −10.84176893776050025102455022923, −9.93135258180035853235257879460, −9.29550620661717767082895875064, −8.9520464631836221624761863557, −8.130478919214785733704828095144, −7.42971308487466675979454019546, −6.462348969578512035138098480291, −5.110047900371747407836044128583, −4.756273428294238385361857976327, −3.844727829778593933851587371567, −3.2382082829640609408890985870, −2.51230764299373766531401139492, −1.76981084663017927294469379377, −0.69356019590185292555831621115, 0.943554742122369956987774076937, 1.45888151528785208686417637611, 2.691205745018441880027223314123, 3.55998659434132808535188518563, 4.23725951981571090073233554230, 5.31876812892304297378438672708, 6.05274449317162135212130993877, 6.61879285271809248339181045697, 7.47132882004514075761935203695, 7.9429706693788862887489926598, 8.68894075773874381456510913702, 9.133739416154025543497045663245, 10.01075651596869326796581537964, 10.73353403727035562084199370590, 11.72189305826512630102450820959, 12.713434665221790527214106248076, 13.23345144684538408002676763841, 13.67274736504370875317648241233, 14.56766469970496263584856333663, 15.0461353351004518968480869876, 15.59509719513258312682946756932, 16.43456253313376341221284750938, 17.13661146033090558358687982791, 17.93124967022581503433623400582, 18.28601670016011856739733706710

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