Properties

Label 1-4033-4033.11-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.668 - 0.743i$
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  + (−0.642 − 0.766i)2-s + (−0.686 − 0.727i)3-s + (−0.173 + 0.984i)4-s + (−0.893 − 0.448i)5-s + (−0.116 + 0.993i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.549 − 0.835i)13-s + (0.957 + 0.286i)14-s + (0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.686 − 0.727i)3-s + (−0.173 + 0.984i)4-s + (−0.893 − 0.448i)5-s + (−0.116 + 0.993i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.549 − 0.835i)13-s + (0.957 + 0.286i)14-s + (0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3371689765 - 0.1503612997i\)
\(L(\frac12)\) \(\approx\) \(0.3371689765 - 0.1503612997i\)
\(L(1)\) \(\approx\) \(0.3519299645 - 0.2588685398i\)
\(L(1)\) \(\approx\) \(0.3519299645 - 0.2588685398i\)

Euler product

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

−18.36238491135534438587369524136, −17.53056543180055062869325936219, −16.83581383966959819522713336762, −16.39796804766929991293158197025, −15.84842254852439890047571900466, −15.11628335778832483864032292411, −14.71034262991048370089048220583, −13.891619586134129319171559230678, −12.89741635274521397314330578246, −11.84519779525631617562459102927, −11.53577115185241184504679833385, −10.54978008058328060362320722145, −10.04975984979671448759010384871, −9.41431142035902888944188548576, −8.88891292040645917306719028531, −7.590257569432735117321673711322, −7.05649800699326436084508678770, −6.78992361019883228062539156061, −5.7257793770944836542466058603, −4.98819644135611929368771172139, −4.14602046181389538654251285785, −3.7052102502481890293346918031, −2.438818780814837795406681502155, −1.13657880791892010863459322306, −0.19783561905689391743251654988, 0.462390140593531806576929031386, 0.99254151395313915710920651653, 2.17531881462532333668343132981, 2.9742845813169901207586171977, 3.627656616835353446296901272701, 4.68936109435722760631480734225, 5.42419691033448414450950317041, 6.42951360084668634055781179017, 7.06028560255843662706434747621, 7.89811144731342401744737831419, 8.50871011501050216149558465878, 9.03577250330854143297729834765, 10.04539160216804122791203892547, 10.849776550557974170239819194478, 11.32531460243451766562754219726, 12.04718403899697133525512182800, 12.59026050238869531187322872156, 13.04868963301112221707777732186, 13.65764767687256998864952047947, 15.05030845027025864078456182938, 15.695773720272237645647737080356, 16.584983338709628823990361835821, 16.76739000568878322997625244849, 17.65399196656444067266861418893, 18.32511323617684600785549043992

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