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Properties

Label 1-4033-4033.967-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.958 - 0.286i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.5 + 0.866i)2-s + (−0.396 + 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.957 − 0.286i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s + 8-s + (−0.686 − 0.727i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.993 − 0.116i)12-s + (−0.286 − 0.957i)13-s + (−0.286 − 0.957i)14-s + (−0.116 + 0.993i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.396 + 0.918i)3-s + (−0.5 − 0.866i)4-s + (0.957 − 0.286i)5-s + (−0.597 − 0.802i)6-s + (−0.686 + 0.727i)7-s + 8-s + (−0.686 − 0.727i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.993 − 0.116i)12-s + (−0.286 − 0.957i)13-s + (−0.286 − 0.957i)14-s + (−0.116 + 0.993i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (967, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.958 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1158948189 + 0.7922678761i\)
\(L(\frac12)\) \(\approx\) \(-0.1158948189 + 0.7922678761i\)
\(L(1)\) \(\approx\) \(0.5156421791 + 0.4883257617i\)
\(L(1)\) \(\approx\) \(0.5156421791 + 0.4883257617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (0.957 - 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.230 + 0.973i)T \)
13 \( 1 + (-0.286 - 0.957i)T \)
17 \( 1 - T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.957 - 0.286i)T \)
31 \( 1 + (-0.448 + 0.893i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (0.918 + 0.396i)T \)
59 \( 1 + (-0.993 - 0.116i)T \)
61 \( 1 + (-0.998 - 0.0581i)T \)
67 \( 1 + (0.116 - 0.993i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (0.835 + 0.549i)T \)
89 \( 1 + (-0.957 + 0.286i)T \)
97 \( 1 + (0.448 + 0.893i)T \)
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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

−18.13429461265904793872642734208, −17.53123799173075409720070528504, −16.941816780779179621465826902596, −16.573842046768247351733697567895, −15.572222883476092244856345350330, −13.98298330735558864213238990609, −13.84359055056524651396376530423, −13.35701251833991923219116339828, −12.63448735207335590700812181419, −11.81629113679405217925794629666, −11.10263447983424158715576796547, −10.72799243472379422989145450552, −9.82783482500829015391392070278, −9.08803068211119403428451490718, −8.646999729468413538323293800820, −7.36713119639035329346832224359, −7.017443366088759281328630192317, −6.28382411209908066082650009379, −5.449994850975920832234849247990, −4.429739750871816293692516373271, −3.515551329676787386020389582608, −2.61728729423225104201579236913, −2.073743857270651451168930916242, −1.163862100144163188876854614279, −0.3462702332290826905286528828, 0.96609553858732436823204740571, 2.13263884327667706296786676787, 2.96126474645160125076988446491, 4.2501654492261665996733772555, 4.86821253161907448397644700197, 5.48723580095675677350312677883, 6.34913540806478990944424144280, 6.453021919903613764365767326279, 7.7649706905888394529345730164, 8.63606574754747652268503185129, 9.24763439385026488468527824913, 9.704732630850917035466278020892, 10.34673833369165486434588557648, 10.80705769695882012944027157129, 12.24933933002216927328807651038, 12.55075953442515100425046104255, 13.544943974236900057853683988384, 14.40142887833874724421904441386, 14.93565801246669149766370617161, 15.5637410297327477799216946022, 16.16310585124785138651105179774, 16.763743394228355913754346721094, 17.41295003564078146305927144090, 18.05459341517483194202945591453, 18.31315856808788252476954770723

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