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Properties

Label 1-4033-4033.1317-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.0639 - 0.997i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  − i·2-s + (−0.973 + 0.230i)3-s − 4-s + (0.998 − 0.0581i)5-s + (0.230 + 0.973i)6-s + (0.893 + 0.448i)7-s + i·8-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.973 − 0.230i)12-s + (0.998 − 0.0581i)13-s + (0.448 − 0.893i)14-s + (−0.957 + 0.286i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (−0.973 + 0.230i)3-s − 4-s + (0.998 − 0.0581i)5-s + (0.230 + 0.973i)6-s + (0.893 + 0.448i)7-s + i·8-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.973 − 0.230i)12-s + (0.998 − 0.0581i)13-s + (0.448 − 0.893i)14-s + (−0.957 + 0.286i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0639 - 0.997i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.0639 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.571192120 - 1.675021204i\)
\(L(\frac12)\) \(\approx\) \(1.571192120 - 1.675021204i\)
\(L(1)\) \(\approx\) \(0.9268614611 - 0.4767830750i\)
\(L(1)\) \(\approx\) \(0.9268614611 - 0.4767830750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.998 - 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (0.998 - 0.0581i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.448 - 0.893i)T \)
31 \( 1 + (0.116 + 0.993i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (-0.686 - 0.727i)T \)
59 \( 1 + (0.957 - 0.286i)T \)
61 \( 1 + (-0.802 - 0.597i)T \)
67 \( 1 + (-0.973 - 0.230i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.893 + 0.448i)T \)
79 \( 1 + (0.957 - 0.286i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (0.549 + 0.835i)T \)
97 \( 1 + (0.116 - 0.993i)T \)
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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

−18.02968838235390423285367478975, −17.60688431422365162751832710852, −17.33159279516279934192592112843, −16.65892525608236507578688273285, −15.72976300675639116374144733655, −15.240217715182212387640575212560, −14.40566062500785629751173710558, −13.62050262464469723082883307122, −13.227321835124067133881275669127, −12.546249391873017467942704754265, −11.584616301457335218919288410014, −10.60633604376641311963282577905, −10.43487229386577328122861683064, −9.27581407371192080728534314116, −8.80111885782426836844170593918, −7.691724185085042641561224732379, −7.18089754061562896285275285550, −6.41400574311934659414289735984, −5.91949355839994119432201198761, −5.15263133795197293545117247253, −4.48578842785982482618437027774, −3.94699436612024529642822407683, −2.23062467258520920880944408898, −1.4925876793712739783548665327, −0.72005637888068922244610677179, 0.557393988257541604737239070119, 1.21952050923751310446357999658, 2.009563843762212720667415503955, 2.79673395073378322336555846954, 3.92568464532883068973015758098, 4.60298932778598991816389740874, 5.26390051720368328312466695922, 6.08878703384432037336901131313, 6.34838273435482135764167396698, 7.921992674083886620945676465705, 8.66907433177331306743747723282, 9.167570318848129182340843463444, 10.07961300730573650045188662932, 10.80319165296953706375862718361, 11.04297281097529350690078211418, 11.81327901122406942462107627066, 12.555404703128115408592711415632, 13.15213392310914096754107491576, 13.99736716416129554904486885212, 14.37022022881151204314407353985, 15.47280350375263761942091784554, 16.29670590889162023076227282970, 16.946240985910271001747925584277, 17.81799192450578306611476503966, 17.96144277290978402194630200114

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