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Properties

Label 1-4235-4235.1784-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.779 + 0.626i$
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  + (0.198 − 0.980i)2-s + (0.309 − 0.951i)3-s + (−0.921 − 0.389i)4-s + (−0.870 − 0.491i)6-s + (−0.564 + 0.825i)8-s + (−0.809 − 0.587i)9-s + (−0.654 + 0.755i)12-s + (−0.974 + 0.226i)13-s + (0.696 + 0.717i)16-s + (0.362 − 0.931i)17-s + (−0.736 + 0.676i)18-s + (0.774 − 0.633i)19-s + (0.142 + 0.989i)23-s + (0.610 + 0.791i)24-s + (0.0285 + 0.999i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.198 − 0.980i)2-s + (0.309 − 0.951i)3-s + (−0.921 − 0.389i)4-s + (−0.870 − 0.491i)6-s + (−0.564 + 0.825i)8-s + (−0.809 − 0.587i)9-s + (−0.654 + 0.755i)12-s + (−0.974 + 0.226i)13-s + (0.696 + 0.717i)16-s + (0.362 − 0.931i)17-s + (−0.736 + 0.676i)18-s + (0.774 − 0.633i)19-s + (0.142 + 0.989i)23-s + (0.610 + 0.791i)24-s + (0.0285 + 0.999i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.779 + 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1265874856 - 0.04461133123i\)
\(L(\frac12)\) \(\approx\) \(-0.1265874856 - 0.04461133123i\)
\(L(1)\) \(\approx\) \(0.5670776837 - 0.7022001506i\)
\(L(1)\) \(\approx\) \(0.5670776837 - 0.7022001506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.198 - 0.980i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.974 + 0.226i)T \)
17 \( 1 + (0.362 - 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.254 - 0.967i)T \)
31 \( 1 + (-0.0855 - 0.996i)T \)
37 \( 1 + (-0.516 + 0.856i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.736 - 0.676i)T \)
53 \( 1 + (-0.696 + 0.717i)T \)
59 \( 1 + (-0.993 - 0.113i)T \)
61 \( 1 + (0.198 + 0.980i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (-0.897 + 0.441i)T \)
79 \( 1 + (-0.941 - 0.336i)T \)
83 \( 1 + (0.466 + 0.884i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.610 + 0.791i)T \)
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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

−19.02007529119734077100372605044, −17.98331823256934690849305580540, −17.49792945216031826468488968504, −16.63423674516791382231107383765, −16.30372932234826962709343414191, −15.63954866673130294043822620084, −14.80511776460078053148371344733, −14.43064100708235505961576786890, −14.01906370453990799658187036663, −12.79630507789676133148924518939, −12.490863305451525262714343861887, −11.455817840095846371978546438, −10.378564600688890789511919266486, −10.05003257761693292461993715820, −9.17999841602011579497152243181, −8.58551683374411432870249322494, −7.90117229573175222574722019110, −7.24146279924597044891490630292, −6.2838510302565632923246897527, −5.547490154822216873130509157208, −4.89738118908068914207566827571, −4.32609177129051219187886399901, −3.37247463114778525783212723229, −2.9236970242376272499766235145, −1.54576406835120538143380752332, 0.035283607979499467167034083405, 1.03944228128673677002909791512, 1.8129185523736181985904118933, 2.71761142211277848473343601008, 3.064571553263886445638631237944, 4.12714073914039439130353110594, 5.03105657907917142058261551013, 5.63600516879150204460286969755, 6.599928090265906689263462377896, 7.48015531203209954968284265141, 7.94922849109748477243533999296, 9.02584971541709836461745454466, 9.480105768331426747425672894971, 10.11566474675306484399233171867, 11.263800345944135902979492558201, 11.78197865180917267181174575078, 12.114337184183462911976389276115, 13.088540645368142395993666012062, 13.57153491478553144908011689672, 14.05565701223170223543880247255, 14.84821329746536312980653416511, 15.46987034407975716311939814125, 16.68505002497852493173820794289, 17.42237051188843696491300953800, 17.89802326117856934982661388968

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